BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220106T153000Z
DTEND;VALUE=DATE-TIME:20220106T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/1
DESCRIPTION:Title: Wh
at groups? What representations? What software?\nby David Vogan (MIT)
as part of Real reductive groups/atlas\n\n\nAbstract\nThe atlas software i
s about a connected reductive algebraic group $G$ over the field ${\\mathb
b R}$ of real numbers\, with group of real points $G({\\mathbb R})$. It is
possible to tell the software to consider ANY such $G({\\mathbb R})$\, of
rank up to 32\, subject to memory size limitations in the computer. (The
distinction between $G$ and $G({\\mathbb R})$ is of fundamental theoretica
l importance\, but inside the software we will often ignore it.) In practi
ce\, the software recognizes many more or less standard names for importan
t examples of real $G({\\mathbb R})$\, and one can just learn to use those
.\n
LOCATION:https://researchseminars.org/talk/atlas/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220113T153000Z
DTEND;VALUE=DATE-TIME:20220113T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/2
DESCRIPTION:Title: Ro
ot data: how atlas understands a reductive group\nby Jeffrey Adams (Un
iversity of Maryland) as part of Real reductive groups/atlas\n\n\nAbstract
\nA root datum is a discrete mathematics object which (by the work of Cart
an-Killing\, Chevalley\, and Grothendieck) can be used to specify a reduct
ive algebraic group $G$ over any algebraically closed field $k$. This is t
he starting point for how the ${\\tt atlas}$ software can work with a real
reductive group.\n
LOCATION:https://researchseminars.org/talk/atlas/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220120T153000Z
DTEND;VALUE=DATE-TIME:20220120T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/3
DESCRIPTION:Title: Pa
rameters: how atlas understands a representation\nby David Vogan (MIT)
as part of Real reductive groups/atlas\n\n\nAbstract\nThe representation
theory of a real reductive group $G$ is parallel to the theory of Verma mo
dules. There is a (concrete\, not too complicated) set of ``Langlands para
meters" for $G$. For each Langlands parameter there is a standard represen
tation $I(p)$\, analogous to a Verma module: (relatively) easy to construc
t and understand. Each standard representation has a unique irreducible (L
anglands) quotient representation $J(p)$\, which can be a small and subtle
part of $I(p)$. Listing irreducible representations is therefore fairly e
asy\; understanding them is harder.\n
LOCATION:https://researchseminars.org/talk/atlas/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220127T153000Z
DTEND;VALUE=DATE-TIME:20220127T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/4
DESCRIPTION:Title: Br
anching to $K$: writing an infinite-dimensional representation of $G$ as a
sum of finite-dimensionals of $K$\nby Jeffrey Adams (University of Ma
ryland) as part of Real reductive groups/atlas\n\n\nAbstract\nThis is a to
pic in which we can compute almost anything\, but where our understanding
of the answers is much more limited. That makes it an excellent topic for
research!\n
LOCATION:https://researchseminars.org/talk/atlas/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220203T153000Z
DTEND;VALUE=DATE-TIME:20220203T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/5
DESCRIPTION:Title: Un
derstanding $K$: how atlas understands Cartan's theory of maximal compact
subgroups\nby David Vogan (MIT) as part of Real reductive groups/atlas
\n\n\nAbstract\nTo study an algebraic variety $X$ over a finite field ${\\
mathbb F}_q$\, we use the Frobenius map $F$\, an algebraic map from $X$ to
$X$ whose set of fixed points is precisely $X({\\mathbb F}_q)$. The power
of this tool stems from the fact that $F$ (unlike the Galois group action
) is algebraic.\n\nTo study a reductive algebraic group $G$ over ${\\mathb
b R}$\, Cartan introduced the Cartan involution $\\theta$\, an algebraic a
utomorphism of $G$ having order 2. The group of fixed points of $\\theta$
is an algebraic subgroup $K$ of $G$. It is not true (as in the finite fiel
d case) that $K$ is equal to $G({\\mathbb R})$\, but Cartan showed how to
get a great deal of information about $G({\\mathbb R})$ from $K$. This ide
a is at the heart of almost all that ${\\tt atlas}$ does.\n
LOCATION:https://researchseminars.org/talk/atlas/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220210T153000Z
DTEND;VALUE=DATE-TIME:20220210T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/6
DESCRIPTION:Title: Th
e Atlas Way (More on KGB)\nby Jeffrey Adams (University of Maryland) a
s part of Real reductive groups/atlas\n\n\nAbstract\nExplain the notion of
"strong real form\," which is central to how atlas writes down Langlands
parameters.\n
LOCATION:https://researchseminars.org/talk/atlas/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220217T153000Z
DTEND;VALUE=DATE-TIME:20220217T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/7
DESCRIPTION:Title: Ch
aracter formulas and Kazhdan-Lusztig polynomials (or why we needed this so
ftware in the first place)\nby David Vogan (MIT) as part of Real reduc
tive groups/atlas\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/atlas/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220224T153000Z
DTEND;VALUE=DATE-TIME:20220224T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/8
DESCRIPTION:Title: Si
gnature character formulas and unitary representations (or why we needed t
his software in the first place)\nby Jeffrey Adams (University of Mary
land) as part of Real reductive groups/atlas\n\n\nAbstract\nWe first discu
ss how it might be possible to write a finite formula for the signature of
a hermitian form on an infinite-dimensional space. Then we explain how at
las computes that signature.\n
LOCATION:https://researchseminars.org/talk/atlas/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220303T153000Z
DTEND;VALUE=DATE-TIME:20220303T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/9
DESCRIPTION:Title: Si
gnature character formulas and unitary representations 2\nby Jeffrey A
dams (University of Maryland) as part of Real reductive groups/atlas\n\n\n
Abstract\nMore details about how atlas computes signatures: realize a repr
esentation as a Langlands quotient J(delta\\otimes nu)\, then deform nu to
zero. At nu=0\, the form is unitary. As nu changes\, the signature of the
form changes only at the (finitely many) values of t*nu (t in [0\,1]) for
which I(delta\\otimes t*nu) is reducible.\n
LOCATION:https://researchseminars.org/talk/atlas/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220317T143000Z
DTEND;VALUE=DATE-TIME:20220317T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/10
DESCRIPTION:Title: W
eyl group representations and atlas\nby Jeffrey Adams (University of M
aryland) as part of Real reductive groups/atlas\n\n\nAbstract\nMuch of the
structure and representation theory of a reductive algebraic group G is d
escribed using the Weyl group W. This talk will introduce what atlas knows
about the structure and representations of W.\n
LOCATION:https://researchseminars.org/talk/atlas/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220331T143000Z
DTEND;VALUE=DATE-TIME:20220331T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/11
DESCRIPTION:Title: G
elfand-Kirillov dimension and atlas\nby David Vogan (MIT) as part of R
eal reductive groups/atlas\n\n\nAbstract\nLinear algebra wants to be about
diagonal matrices. Nilpotent matrices control exactly how this desire mus
t be frustrated\, and what is actually true instead. In a precisely parall
el way\, the theory of reductive groups wants to be about maximal tori. Ni
lpotent classes (for example in the Lie algebra) control how this desire m
ust be frustrated\, and what is actually true instead. The talk will conce
rn what atlas knows about nilpotent Lie algebra elements\, with perhaps so
me small hints about what this information does for representation theory.
\n\nSeminar was actually all about Gelfand-Kirillov dimension: the reason
is that this is a really elementary and interesting invariant that you can
only hope to compute using nilpotent orbits.\n
LOCATION:https://researchseminars.org/talk/atlas/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220310T153000Z
DTEND;VALUE=DATE-TIME:20220310T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/12
DESCRIPTION:Title: H
ow atlas does what it says its doing\nby David Vogan (MIT) as part of
Real reductive groups/atlas\n\n\nAbstract\nOften the descriptions we've sk
etched of the mathematical objects that atlas studies are very close to th
eir internal representations in the software. I'll talk about two situatio
ns where that's not exactly the case: Fokko du Cloux's compact representat
ion of Weyl group elements\, and the replacement of parameters by the "fac
ets" in which they lie.\n
LOCATION:https://researchseminars.org/talk/atlas/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220324T143000Z
DTEND;VALUE=DATE-TIME:20220324T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/13
DESCRIPTION:Title: W
eyl group representations and atlas II\nby Jeffrey Adams (University o
f Maryland) as part of Real reductive groups/atlas\n\n\nAbstract\nContinua
tion of last week's seminar. Particular topics to include branching of rep
resentations from a Weyl group to a subgroup\; motivation of the definitio
n of coherent families\; relationship between Lusztig's "families" in W^ a
nd cell representations of W.\n\nIf time permits\, David Vogan will do the
second half of this seminar\, on the topic of Gelfand-Kirillov dimension
(meant as an introduction to the nilpotent orbit talk next week).\n
LOCATION:https://researchseminars.org/talk/atlas/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220407T143000Z
DTEND;VALUE=DATE-TIME:20220407T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/14
DESCRIPTION:Title: N
ilpotent orbits and atlas\nby David Vogan (MIT) as part of Real reduct
ive groups/atlas\n\n\nAbstract\nLinear algebra wants to be about diagonal
matrices. Nilpotent matrices control exactly how this desire must be frust
rated\, and what is actually true instead. In a precisely parallel way\, t
he theory of reductive groups wants to be about maximal tori. Nilpotent cl
asses (for example in the Lie algebra) control how this desire must be fru
strated\, and what is actually true instead. The talk will concern what at
las knows about nilpotent Lie algebra elements. The goal is to explain how
this helps with the computation of Gelfand-Kirillov dimension.\n
LOCATION:https://researchseminars.org/talk/atlas/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220414T143000Z
DTEND;VALUE=DATE-TIME:20220414T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/15
DESCRIPTION:Title: C
ell representations continued\nby Jeffrey Adams (University of Marylan
d) as part of Real reductive groups/atlas\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/atlas/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220421T143000Z
DTEND;VALUE=DATE-TIME:20220421T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/16
DESCRIPTION:Title: L
usztig's parametrization of families\nby Jeffrey Adams (University of
Maryland) as part of Real reductive groups/atlas\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/atlas/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220428T143000Z
DTEND;VALUE=DATE-TIME:20220428T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/17
DESCRIPTION:Title: R
eal parabolic subgroups and induction in atlas\nby David Vogan (MIT) a
s part of Real reductive groups/atlas\n\n\nAbstract\nThe oldest constructi
on of irreducible unitary representactions of a real reductive group G(R)
is unitary induction from a real parabolic subgroup P(R). A bit more preci
sely\, P(R) has a well-defined normal subgroup U(R)\, the unipotent radica
l\; and the quotient L(R) = P(R)/U(R) is again a real reductive group. "Re
al parabolic induction" means starting with an irreducible unitary represe
ntation pi_L of L(R)\, lifting it to P(R) by making U(R) act trivially\, t
hen applying Mackey induction from P(R) to G(R).\n\nOf course atlas lives
in the rather different world of (g\, K)-modules. I'll explain a bit about
how to translate between these two worlds\, and what atlas can tell you a
bout real parabolic induction.\n
LOCATION:https://researchseminars.org/talk/atlas/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220505T143000Z
DTEND;VALUE=DATE-TIME:20220505T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/18
DESCRIPTION:Title: T
heta-stable parabolic subgroups and cohomological induction in atlas\n
by Jeffrey Adams (University of Maryland) as part of Real reductive groups
/atlas\n\n\nAbstract\nIt has been understood since the 1950s that (unitary
) representations of reductive groups should correspond approximately to (
unitary) characters of Cartan subgroups. Last week we talked about real pa
rabolic induction\, a method also dating to the 1950s for constructing par
t of this correspondence (from the noncompact parts of Cartan subgroups).\
n\nIn the 1970s\, Gregg Zuckerman introduced a parallel method\, called co
homological induction\, for constructing the part of the correspondence c
orresponding to compact parts of Cartan subgroups. We'll explain how that
works\, and how to realize it in the atlas software.\n
LOCATION:https://researchseminars.org/talk/atlas/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220512T143000Z
DTEND;VALUE=DATE-TIME:20220512T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/19
DESCRIPTION:Title: U
nitary dual of SO(2n\,1) in atlas\nby David Vogan (MIT) as part of Rea
l reductive groups/atlas\n\n\nAbstract\nDescription of the unitary dual of
SO(2n\,1) in atlas terms.\n
LOCATION:https://researchseminars.org/talk/atlas/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220519T143000Z
DTEND;VALUE=DATE-TIME:20220519T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/20
DESCRIPTION:Title: U
nitary dual of F4_B4 in atlas\nby David Vogan (MIT) as part of Real re
ductive groups/atlas\n\n\nAbstract\nAbout the Baldoni-Silva/Barbasch class
ification of unitary representations of the rank one form of F4\, as seen
by atlas.\n
LOCATION:https://researchseminars.org/talk/atlas/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220526T143000Z
DTEND;VALUE=DATE-TIME:20220526T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/21
DESCRIPTION:Title: L
oose ends: Hermitian representations\, more on parameters\, translation an
d the Jantzen filtration\nby Jeffrey Adams (University of Maryland) as
part of Real reductive groups/atlas\n\n\nAbstract\nA few miscellaneous to
pics: Hermitian representations and the Hermitian dual\, more about parame
ters\, translation functors. We didn't get to the Jantzen filtration\; to
be continued next week\n
LOCATION:https://researchseminars.org/talk/atlas/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220602T143000Z
DTEND;VALUE=DATE-TIME:20220602T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/22
DESCRIPTION:Title: M
ore loose ends: translation\, Jantzen filtration\nby Jeffrey Adams (Un
iversity of Maryland) as part of Real reductive groups/atlas\n\nAbstract:
TBA\n
LOCATION:https://researchseminars.org/talk/atlas/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220609T143000Z
DTEND;VALUE=DATE-TIME:20220609T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/23
DESCRIPTION:Title: J
antzen filtration and open mic night\nby Jeffrey Adams (University of
Maryland) as part of Real reductive groups/atlas\n\n\nAbstract\nI will fin
ish going over a few recent topic\, and talk about the Jantzen filtration.
Then I'll take questions. If there's anything you'd like to discuss this
is good chance to bring it up.\n
LOCATION:https://researchseminars.org/talk/atlas/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:ONE WEEK BREAK
DTSTART;VALUE=DATE-TIME:20220616T143000Z
DTEND;VALUE=DATE-TIME:20220616T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/24
DESCRIPTION:Title: N
o seminar this Thursday\, resuming next week\nby ONE WEEK BREAK as par
t of Real reductive groups/atlas\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/atlas/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220623T143000Z
DTEND;VALUE=DATE-TIME:20220623T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/25
DESCRIPTION:Title: D
irac operator in atlas\nby David Vogan (MIT) as part of Real reductive
groups/atlas\n\n\nAbstract\nThe Dirac operator was introduced by Parthasa
rathy and others in the 1970s as a tool for studying unitary representatio
ns. One of its most powerful aspects is Parthasarathy's Dirac inequality\,
which provides an upper bound on the infinitesimal character of a unitary
representation containing a particular K-type. I'll explain how to comput
e this by hand\, then introduce an atlas script which does the job. A unit
ary representation is said to _have Dirac cohomology_ if equality holds in
the Dirac inequality\; such representations have been studied extensively
by Barbasch\, Ding\, Dong\, Huang\, Mehdi\, Pandzic\, Wong\, Zierau... an
d I apologize to those omitted. I will show how atlas computes Dirac cohom
ology.\n
LOCATION:https://researchseminars.org/talk/atlas/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220630T143000Z
DTEND;VALUE=DATE-TIME:20220630T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/26
DESCRIPTION:Title: C
lassifying the unitary dual (part 1 of infinitely many...)\nby David V
ogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nWe've exp
lained a lot about how atlas can offer information about individual repres
entations\, and certainly how it can check whether an individual represent
ation is unitary. Want to start talking about how to describe the full uni
tary dual: to give a finite and complete description of the answers to inf
initely many questions "is it unitary?" The Dirac inequality from last wee
k is useful hint\; I will try to say how it can be part of a general pictu
re of the unitary dual\, and how it might usefully be modified.\n
LOCATION:https://researchseminars.org/talk/atlas/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220707T143000Z
DTEND;VALUE=DATE-TIME:20220707T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/27
DESCRIPTION:Title: A
ffine Weyl group facets and the unitary dual\nby David Vogan (MIT) as
part of Real reductive groups/atlas\n\n\nAbstract\nLast week we saw that t
here were two kinds of finiteness problems standing in the way of a finite
description of the unitary dual. Today I'll focus on the second one: divi
ding the continuous parameters of possibly unitary representations into a
finite set of pieces where unitarity is constant. (Don't worry\, Jeff wil
l come back soon!)\n
LOCATION:https://researchseminars.org/talk/atlas/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220721T143000Z
DTEND;VALUE=DATE-TIME:20220721T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/28
DESCRIPTION:Title: V
ogan duality\nby Jeffrey Adams (University of Maryland) as part of Rea
l reductive groups/atlas\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/atlas/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220728T143000Z
DTEND;VALUE=DATE-TIME:20220728T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/29
DESCRIPTION:Title: A
rthur packets\nby Jeffrey Adams (University of Maryland) as part of Re
al reductive groups/atlas\n\n\nAbstract\nCompletion of the discussion of V
ogan duality from last week\; application to the definition and computatio
n of Arthur packets of unipotent representations.\n
LOCATION:https://researchseminars.org/talk/atlas/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:NONE
DTSTART;VALUE=DATE-TIME:20220714T143000Z
DTEND;VALUE=DATE-TIME:20220714T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/30
DESCRIPTION:Title: T
AKE A WEEK OFF\nby NONE as part of Real reductive groups/atlas\n\nAbst
ract: TBA\n
LOCATION:https://researchseminars.org/talk/atlas/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220804T143000Z
DTEND;VALUE=DATE-TIME:20220804T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/31
DESCRIPTION:Title: D
uality for singular and non-integral infinitesimal character\nby David
Vogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nJeffrey
Adams explained in the last two lectures how Vogan duality relates repres
entation of a real group G(R) to representations of a real form G^v(R) of
the Langlands dual group\, in the case of regular integral infinitesimal c
haracter. Today I have three goals:\n 1) to say a few words about how
this representation theory duality can be rephrased as\n\n(reps of G(R))
DUAL TO (algebraic geometry of space of Langlands parameters in ^L G)\n\n
so that it can make sense over other local fields\;\n\n 2) Explain wha
t happens for regular but NON integral infinitesimal character. (Answer: r
eal form of G^v is replaced by real form of pseudolevi subgroup G^v(gamma)
).\n\n 3) Explain what happens for SINGULAR infinitesimal character. (
Answer: translation principle tells you everything\, but what it tells yo
u is a bit complicated.)\n\nThe subgroup G^(gamma) is dual to a Langlands-
Shelstad endoscopic group for G. What we do with G^(gamma) is certainly re
lated to endoscopy\, but it is not at all precisely the same thing.\n
LOCATION:https://researchseminars.org/talk/atlas/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220811T143000Z
DTEND;VALUE=DATE-TIME:20220811T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/32
DESCRIPTION:Title: D
uality for singular integral infinitesimal character\nby David Vogan (
MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nLast week I rep
eated Jeff's description of duality as a bijection.\n\n(reps with regular
integral infl char of real forms of G) <--->\n (reps with regular inte
gral infl char of real forms of G^\\vee)\n\nToday I will start by explaini
ng how this changes for singular integral infl char:\n\n(reps of forms of
G\, integral infl char singular on simple roots S) <--->\n (reps of
forms of G^\\vee\, integral infl char\, simple roots S^\\vee in tau invari
ant)\n\nCondition on the G^vee side is that the reps must be somewhat SMAL
L. A special case is what Jeff discussed already\n\n(forms of G reps\, inf
l char zero) <----> (fin-diml reps for forms of G^\\vee)\n
LOCATION:https://researchseminars.org/talk/atlas/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220818T143000Z
DTEND;VALUE=DATE-TIME:20220818T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/33
DESCRIPTION:Title: H
ermitian forms on finite- dimensional representations\nby Jeffrey Adam
s (University of Maryland) as part of Real reductive groups/atlas\n\n\nAbs
tract\nThe study of signatures of Hermitian forms on finite dimensional re
presentations serves as an example of the general theory\, while having so
me special features\, and being surprisingly interesting in its own right.
\n\nThe finite dimensional representations of the real forms $G(\\mathbb R
)$ of $G(\\mathbb C$) are essentially independent of the real form. Howeve
r these representations are all unitary if and only if $G(\\mathbb R)$ is
compact\, and the signature of the Hermitian forms depend very much on the
real form. \n\nWe'll talk about computing this\, using an elementary form
ula (arising from the Atlas theory of the c-form)in terms of the Weyl char
acter formula. We'll also discuss some interesting invariants for a finite
dimensional representation: the Frobenius/Schur indicator and the real-qu
aternionic indicator.\n
LOCATION:https://researchseminars.org/talk/atlas/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220825T143000Z
DTEND;VALUE=DATE-TIME:20220825T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/34
DESCRIPTION:Title: C
ohomological induction and restricting discrete series to K\nby David
Vogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nHarish-C
handra's theorem (and also the Langlands classification\, which is based o
n the theorem) says that a discrete series representation of a real reduct
ive G is specified by a character of a compact Cartan subgroup T of G. Thi
s talk is about Zuckerman's idea of how to implement that: given a charact
er of a compact Cartan\, how to construct a (g.K) module. The construction
is most explicit as a representation of K\; so I'll talk about how to see
the restriction to K of a discrete series\, and what things we do and don
't know about that.\n
LOCATION:https://researchseminars.org/talk/atlas/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220901T143000Z
DTEND;VALUE=DATE-TIME:20220901T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/35
DESCRIPTION:Title: M
ore about discrete series restriction\nby David Vogan (MIT) as part of
Real reductive groups/atlas\n\n\nAbstract\nLast week I described Zuckerma
n's construction of the (g\,K)-module of a discrete series representation.
This week I'll look at how to make that construction explicit: how to ext
ract the Blattner formula\n
LOCATION:https://researchseminars.org/talk/atlas/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220908T143000Z
DTEND;VALUE=DATE-TIME:20220908T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/36
DESCRIPTION:Title: C
ohomological Arthur packets\nby Jeffrey Adams (University of Maryland)
as part of Real reductive groups/atlas\n\n\nAbstract\nAn important specia
l case of Arthur packets are those of regular integral infinitesimal chara
cter. The trivial representation (attached to the dual principal nilpotent
orbit) is an example. \n\nIt is known by a result of Salamanca that the u
nitary representations with regular integral infinitesimal character are p
recisely the cohomological representations. These are representations with
non-trivial twisted $(\\mathfrak g\,K)$ cohomology. By a result of Vogan
and Zuckerman these are precisely the modules $A_\\mathfrak q(\\lambda)$\,
constructed via cohomological induction from a unitary character of theta
-stable Levi subgroup. \n\nThe conclusion is: assuming all is right with t
he world (i.e. Arthur's conjectures) an Arthur packet consisting of repres
entations with regular integral infinitesimal character\nmust consist of c
ertain $A_\\mathfrak q(\\lambda)$-modules. These are sometimes referred to
as "Adams-Johnson" packets\; these were among the first interesting Arthu
r packets to be studied in the 1980s.\n\nI'll discuss these things in the
context of Atlas.\n
LOCATION:https://researchseminars.org/talk/atlas/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220915T143000Z
DTEND;VALUE=DATE-TIME:20220915T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/37
DESCRIPTION:Title: C
ohomological Arthur Packets 2\nby Jeffrey Adams (University of Marylan
d) as part of Real reductive groups/atlas\n\n\nAbstract\n(This is a contin
uation of the talk from last week)\n\nAn important special case of Arthur
packets are those of regular integral infinitesimal character. The trivial
representation (attached to the dual principal nilpotent orbit) is an exa
mple. \n\nIt is known by a result of Salamanca that the unitary representa
tions with regular integral infinitesimal character are precisely the coho
mological representations. These are representations with non-trivial twis
ted $(\\mathfrak g\,K)$ cohomology. By a result of Vogan and Zuckerman the
se are precisely the modules $A_\\mathfrak q(\\lambda)$\, constructed via
cohomological induction from a unitary character of theta-stable Levi subg
roup. \n\nThe conclusion is: assuming all is right with the world (i.e. Ar
thur's conjectures) an Arthur packet consisting of representations with re
gular integral infinitesimal character\nmust consist of certain $A_\\mathf
rak q(\\lambda)$-modules. These are sometimes referred to as "Adams-Johnso
n" packets\; these were among the first interesting Arthur packets to be s
tudied in the 1980s.\n\nI'll discuss these things in the context of Atlas.
\n
LOCATION:https://researchseminars.org/talk/atlas/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220922T143000Z
DTEND;VALUE=DATE-TIME:20220922T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/38
DESCRIPTION:Title: C
ohomological Arthur packets 3/Open Mic NightI'l\nby Jeffrey Adams (Uni
versity of Maryland) as part of Real reductive groups/atlas\n\n\nAbstract\
nWe'll finish our discussion of cohomological Arthur packets (see the prev
ious talk for an abstract). This should leave time for questions\, about a
nything at all. If you have an example you'd like to see worked out in atl
as\, be prepared to ask about it.\n
LOCATION:https://researchseminars.org/talk/atlas/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20220929T143000Z
DTEND;VALUE=DATE-TIME:20220929T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/39
DESCRIPTION:Title: D
uality\, associated varieties\, and nilpotent orbits\nby David Vogan (
MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nAt the beginnin
g of August I talked about "duality\," which relates a category of (g\,K)-
modules to a geometric category on the Langlands dual group G^\\vee. (I co
uld have said "to a category of (g^\\vee\, K^\\vee)-modules" but the formu
lation above works for p-adic G as well.) Today I'll look at associated va
rieties in this context\, recovering various notions of "duality" for nilp
otent orbits (due originally to Spaltenstein\, Lusztig\, and Barbasch-Voga
n).\n
LOCATION:https://researchseminars.org/talk/atlas/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:NONE
DTSTART;VALUE=DATE-TIME:20221006T143000Z
DTEND;VALUE=DATE-TIME:20221006T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/40
DESCRIPTION:Title: N
O MEETING THIS WEEK!\nby NONE as part of Real reductive groups/atlas\n
\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/atlas/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20221013T143000Z
DTEND;VALUE=DATE-TIME:20221013T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/41
DESCRIPTION:Title: D
uality for G reps\, nilpotent orbits\, and W reps\nby David Vogan (MIT
) as part of Real reductive groups/atlas\n\n\nAbstract\nI will try to summ
arize all the kinds of duality we've talked about: for HC modules\, for ni
lpotent orbits\, and for W representations. I'll try to say what they have
to do with each other\, and mention open problems about them.\n\nUndersta
nding all of this appropriately seems to be a way to understand associated
varieties of Harish-Chandra modules.\n
LOCATION:https://researchseminars.org/talk/atlas/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20221020T143000Z
DTEND;VALUE=DATE-TIME:20221020T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/42
DESCRIPTION:Title: D
uality for G reps\, nilpotent orbits\, and W reps III\nby David Vogan
(MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nLast week I ta
lked in some detail about duality in the case of nonintegral infinitesimal
character. This involved a pseudoLevi subgroup D^\\vee of G^vee (centrali
zer of exponential of infinitesimal character)\, and a corresponding endos
copic group D sharing the Cartan H of G\; roots of D are the gamma-integra
l roots. \n\nDUALITY relates reps of G of infl char gamma and reps of D^\\
vee. Therefore there is an EQUIVALENCE between reps of G of infl char gamm
a and reps of D of (D-integral) infl char gamma). Today I'll talk about ho
w (complex) nilpotent orbits and W rep move in this equivalence. I will tr
y to describe a great research topic: to understand how REAL nilpotent orb
its move between D and G.\n
LOCATION:https://researchseminars.org/talk/atlas/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20221027T143000Z
DTEND;VALUE=DATE-TIME:20221027T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/43
DESCRIPTION:Title: E
xamples of Duality/Miscellaneous\nby Jeffrey Adams (University of Mary
land) as part of Real reductive groups/atlas\n\n\nAbstract\nI'll do some a
tlas examples illustrating the duality theory of the past few weeks.\nThen
I'll cover a few other topics which I've been asked about recently. Possi
bilities include: computing derived functors outside of the good range\, d
etails about unipotent representations of $G_2$\, large representations an
d Whittaker models.\n
LOCATION:https://researchseminars.org/talk/atlas/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20221103T143000Z
DTEND;VALUE=DATE-TIME:20221103T160000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/44
DESCRIPTION:Title: M
ore on Duality/Miscellaneous\nby Jeffrey Adams (University of Maryland
) as part of Real reductive groups/atlas\n\n\nAbstract\nThis is a continua
tion of last week. I'll give a few more atlas examples of duality.\n\nLast
week I defined a correspondence relating certain (special) nilpotent K-or
bits for G to certain (special) nilpotent K^\\vee orbits for G^\\vee. I'll
use atlas to compute some examples of this correspondence.\n\nI'll also t
alk about the Speh representation\, and computin degenerate derived functo
r modules in atlas.\n\nTime permitting: Arthur packets for G2.\n
LOCATION:https://researchseminars.org/talk/atlas/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20221110T153000Z
DTEND;VALUE=DATE-TIME:20221110T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/45
DESCRIPTION:Title: A
rthur packets for G2\nby Jeffrey Adams (University of Maryland) as par
t of Real reductive groups/atlas\n\n\nAbstract\nWe've talked about how to
compute WEAK Arthur packets for general real reductive G: the union over a
ll parameters with a fixed restriction to SL(2) of the Arthur packet. Toda
y I'll look at precisely how these weak packets break into actual packets.
This is subtle (and interesting!) and atlas cannot efficiently compute it
for general G.\n
LOCATION:https://researchseminars.org/talk/atlas/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:NONE
DTSTART;VALUE=DATE-TIME:20221124T153000Z
DTEND;VALUE=DATE-TIME:20221124T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/46
DESCRIPTION:Title: N
O MEETING: US Thanksgiving Holiday\nby NONE as part of Real reductive
groups/atlas\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/atlas/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams\, David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20221117T153000Z
DTEND;VALUE=DATE-TIME:20221117T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/47
DESCRIPTION:Title: M
ore about Arthur packets\nby Jeffrey Adams\, David Vogan (MIT) as part
of Real reductive groups/atlas\n\n\nAbstract\nBegin with Jeffrey talking
about how a weak unipotent Arthur packet for G2 contains two non-unipotent
Arthur packets (as constructed by Adams and Johnson in 1987). These stran
ge overlaps account for the extra stable sums of irreducibles in the weak
packet\, which were displayed last week.\n\nProbably David will then talk
about this: if the SL(2) portion of an Arthur parameter psi is not disting
uished in the dual group\, then the Bala-Carter Levi L^vee for the SL(2) i
s a PROPER Levi. Of course a Levi L^\\vee in G^\\vee corresponds to a Levi
L in G\, and this Levi comes with an inner class of rational forms (altho
ugh it need NOT be the Levi of a rational parabolic). The Arthur parameter
psi for G can be regarded as an Arthur parameter for L. In the real case\
, there are well-behaved "cohomological induction functors" carrying repre
sentations of (these rational forms of) L to representations of (our inner
class of) rational forms of G. \n\nTHESE INDUCTION FUNCTORS CARRY THE ART
HUR PACKET Pi_psi(L) INTO THE ARTHUR PACKET Pi_psi(G).\n\nCONJECTURE: they
are ONTO.\n\nIf this is true\, then Arthur packets are only difficult whe
n the SL(2) part is DISTINGUISHED\; that is\; the nilpotent is rather larg
e\; that is\; the representations are rather far from tempered.\n
LOCATION:https://researchseminars.org/talk/atlas/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20221201T153000Z
DTEND;VALUE=DATE-TIME:20221201T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/48
DESCRIPTION:Title: C
omputing honest Arthur packets\nby David Vogan (MIT) as part of Real r
eductive groups/atlas\n\n\nAbstract\nLong general talk about Langlands' ap
proach to automorphic forms\, and how it leads toward Arthur's conjectures
. Brief demonstration of Annegret Paul's new script\, which actually calcu
lates Arthur packets in many cases.\n
LOCATION:https://researchseminars.org/talk/atlas/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20221208T153000Z
DTEND;VALUE=DATE-TIME:20221208T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/49
DESCRIPTION:Title: A
nnegret Paul's magic script\nby David Vogan (MIT) as part of Real redu
ctive groups/atlas\n\n\nAbstract\nA unipotent Arthur packet is attached to
a nilpotent class in the dual group\, and a finite amount of additional d
ata. A weak pack is the union of all packets attached to a single nilpoten
t class\; the software has been able to compute those (with great effort!)
for a long time. Annegret Paul has a new script which can (conjecturally)
compute actual Arthur packets for those nilpotents in the dual group whic
h are principal in some Levi. Often this is most nilpotents\; for E8\, it
is 41 of the 70 nilpotent classes.\n\nI will recall the theoretical conjec
ture required to PROVE this (without it\, we know only that the script is
producing _part of_ an honest Arthur packet)\; then spend most of the time
looking carefully at how the script works. This is meant to be an exercis
e both in theory and in scripting for atlas.\n
LOCATION:https://researchseminars.org/talk/atlas/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams\, David Vogan (University of Maryland)
DTSTART;VALUE=DATE-TIME:20221215T153000Z
DTEND;VALUE=DATE-TIME:20221215T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/50
DESCRIPTION:Title: I
nteresting examples of Arthur packets\, computed by Annegret's script\
nby Jeffrey Adams\, David Vogan (University of Maryland) as part of Real r
eductive groups/atlas\n\n\nAbstract\nLast time we looked a bit at the scri
pt "test_non_distinguished.at"\, which computes (some) Arthur packets. Thi
s time will emphasize more the interesting examples\, and less the (also i
nteresting!) details about scripting.\n\nThis will be the LAST seminar of
the fall semester. Probably we will not meet again at least until Thursday
\, January 11\; from that time on the schedule is still under discussion.\
n
LOCATION:https://researchseminars.org/talk/atlas/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20230112T153000Z
DTEND;VALUE=DATE-TIME:20230112T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/51
DESCRIPTION:Title: C
omputing unitary duals\, I: cohomological induction\nby David Vogan (M
IT) as part of Real reductive groups/atlas\n\n\nAbstract\nWe are planning
to conclude this seminar (for now?) with three talks January 12\, 19\, and
26\, 2023. Topic is progress and plans for the original goal of the atlas
project: to make software that can describe the unitary dual of any real
reductive group G(R).\n\nThere are two fundamental classical techniques to
construct unitary representations. The first (due to Israel Gelfand and h
is collaborators) is real parabolic induction. A theorem in Knapp's "Overv
iew" book gives a very simple way to identify most of the unitary represen
tations that can be obtained in this way: they are the ones with non-real
infinitesimal character. In light of that theorem\, one can study _only_ r
epresentations with REAL infinitesimal character.\n\nThe second classical
technique (due to Gregg Zuckerman and those who stole from him) is cohomol
ogical parabolic induction. The analogue of the theorem in Knapp's book wo
uld say that any unitary representation with non-imaginary infinitesimal c
haracter can be obtained by cohomological induction. THIS IS NOT TRUE\, bu
t it is nearly true.\n\nWhat's actually true is that any unitary represent
ation for which the real part of the infinitesimal character is LARGE ENOU
GH can be obtained by cohomological induction. The question of what "large
enough" means is best expressed in terms of "nonunitarity certificates. T
oday I will state these results with some care\, and start to look at nonu
nitarity certificates.\n
LOCATION:https://researchseminars.org/talk/atlas/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20230119T153000Z
DTEND;VALUE=DATE-TIME:20230119T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/52
DESCRIPTION:Title: C
omputing unitary duals\, II: nonunitarity certificates\nby David Vogan
(MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nLast week I s
tated the Langlands classification in terms of atlas parameters. I showed
how an atlas parameter defines a real parabolic P^u = M^u N^u\, and stated
the theorem from Knapp's "Overview" that the corresponding representation
pi_G is unitary if and only if it's unitarily induced from pi_{M^u}. Clas
sifying the unitary dual therefore reduces to the case M^u = G\, which is
equivalent to REAL INFINITESIMAL CHARACTER.\n\nNext\, I showed how to atta
ch to an atlas parameter p a theta-stable parabolic q = l+u\, and a parame
ter pL for L\; and sketched how the G-rep pi attached to p is cohomologica
lly induced from the L-rep piL attached to pL.\n\nTopic for today is to un
derstand how the correspondence piL --> pi is related to unitarity.\n\nApp
arently unrelated\, but actually deeply connected: if pi is an irreducible
representation of a real reductive G(R)\, we get a multiplicity function
m_{pi}: K^ --> N\, m_{pi}(mu) = multiplicity of mu in pi|_K. The atlas so
ftware computes this function (up to K-types of any specified height). If
pi admits an invariant Hermitian form\, then the form has a signature: m_p
i is the sum of two N-valued functions p_{pi} and q_{pi}. We arrange the f
orm to be positive definite on some lowest K-type mu_0\, so that p_{pi}(mu
_0) = 1 and q_{pi}(\\mu_0) = 0.\n\nA NONUNITARITY CERTIFICATE for pi is a
particular K-type mu' with the property that q_{pi}(mu') > 0.\n\nFor each
K-type mu there is a FINITE set {mu'_0\,...\,mu'_r} of K-types so that ANY
NONUNITARY REP OF LKT \\mu MUST HAVE A NONUNITARITY CERTIFICATE IN {mu'_i
}. Such a set is called a NONUNITARITY CERTIFICATE FOR THE LOWEST K-TYPE m
u. I will explain how atlas can hope to compute this finite set (it doesn'
t yet!) and how knowledge of such sets can help to compute unitary duals.\
n
LOCATION:https://researchseminars.org/talk/atlas/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20230126T153000Z
DTEND;VALUE=DATE-TIME:20230126T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/53
DESCRIPTION:Title: C
omputing unitary duals\, III: unitary_dual@RealForm\nby David Vogan (M
IT) as part of Real reductive groups/atlas\n\n\nAbstract\nContent today wi
ll depend on what I did or didn't manage to do on January 12 and 19. Appro
ximately the goal is to describe what a unitary-dual-computer in atlas mig
ht look like.\n
LOCATION:https://researchseminars.org/talk/atlas/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20230202T153000Z
DTEND;VALUE=DATE-TIME:20230202T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/54
DESCRIPTION:Title: E
xamples/Open Mic\nby Jeffrey Adams (University of Maryland) as part of
Real reductive groups/atlas\n\n\nAbstract\nI will do some examples of wha
t David Vogan covered the last few weeks. \n\nI will leave time to answer
questions\, on anything atlas-related. Feel free to ask about anything fro
m issues using the software\, to research level questions. I expect to be
able\nto answer most questions of the first type\, and maybe think of an i
nteresting atlas example\nto address the second.\n
LOCATION:https://researchseminars.org/talk/atlas/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20230209T153000Z
DTEND;VALUE=DATE-TIME:20230209T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/55
DESCRIPTION:Title: Q
uestion and Answer session\nby Jeffrey Adams (University of Maryland)
as part of Real reductive groups/atlas\n\n\nAbstract\nThe seminar proper h
as concluded for the time being. However I'll be available to answer quest
ions about the software\, the math behind it\, and whether pigs have wings
.\n
LOCATION:https://researchseminars.org/talk/atlas/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Vogan (MIT)
DTSTART;VALUE=DATE-TIME:20230216T153000Z
DTEND;VALUE=DATE-TIME:20230216T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/56
DESCRIPTION:Title: q
uestion/answer session\nby David Vogan (MIT) as part of Real reductive
groups/atlas\n\n\nAbstract\nQuestions mostly concerned interpreting atlas
K-types and inducing data for representations. The "slides" link is trans
cript of the atlas session.\n
LOCATION:https://researchseminars.org/talk/atlas/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Adams (University of Maryland)
DTSTART;VALUE=DATE-TIME:20230223T153000Z
DTEND;VALUE=DATE-TIME:20230223T170000Z
DTSTAMP;VALUE=DATE-TIME:20241016T073902Z
UID:atlas/57
DESCRIPTION:Title: Q
&A on linear algebra in atlas\nby Jeffrey Adams (University of Marylan
d) as part of Real reductive groups/atlas\n\n\nAbstract\nJeff discussed so
lving linear equations in atlas\, computing weight multiplicities\, and so
me related mathematics.\n
LOCATION:https://researchseminars.org/talk/atlas/57/
END:VEVENT
END:VCALENDAR