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SUMMARY:Ayan Maiti (Oklahoma State Univ.)
DTSTART:20211130T210000Z
DTEND:20211130T220000Z
DTSTAMP:20260423T041503Z
UID:UAANTS/32
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UAANTS/32/">
 WEYL’S LAW FOR CUSP FORMS OF ARBITRARY $K_{\\infty}$-TYPE</a>\nby Ayan M
 aiti (Oklahoma State Univ.) as part of University of Arizona Algebra and N
 umber Theory Seminar\n\n\nAbstract\nLet $M$ be a compact Riemannian manifo
 ld. It was proved by Weyl that number of\nLaplacian eigenvalues less than 
 $T$\, is asymptotic to $C(M)T^{dim(M)/2}$\, where $C(M)$ is the\nproduct o
 f the volume of $M$\, volume of the unit ball and $(2π)^{−dim(M)}$. Let
  $\\Gamma$ be an\narithmetic subgroup of $SL_2(\\mathbb{Z})$ and \\mathbb{
 H}^2 be an upper-half plane. When $M = \\Gamma \\backslash \\mathbb{H}^2$\
 , Weyl’sasymptotic holds true for the discrete spectrum of Laplacian. It
  was proved by Selberg\, who used his celebrated trace formula.\nLet $G$ b
 e a semisimple algebraic group of Adjoint and split type over $\\mathbb{Q}
 $. Let $G(\\mathbb{R})$ be\nthe set of $\\mathbb{R}$-points of $G$. For si
 mplicity of this exposition let us assume that $\\Gamma \\subset G(\\mathb
 b{R})$ be an torsion free arithmetic subgroup. Let $K_{\\infty}$ be the ma
 ximal compact subgroup.\nLet $L^2(\\Gamma \\backslash G(\\mathbb{R})$ be s
 pace of square integrable $\\Gamma$ invariant functions on $G(\\mathbb{R})
 $. Let $L^2_{cusp}(\\Gamma \\backslash G(\\mathbb{R})$ be the cuspidal sub
 space. Let $M = \\Gamma \\backslash G(\\mathbb{R})/K_{\\infty}$ be a local
 ly symmetric space. Suppose $d = dim(\\Gamma \\backslash G(\\mathbb{R})/K_
 {\\infty})$. Then it was proved by Lindenstrauss and Venkatesh\,\nthat num
 ber of spherical\, i.e. bi-$K_{\\infty}$ invariant cuspidal Laplacian eige
 nfunctions\, whose\neigenvalues are less than T is asymptotic to $C(M)T^{d
 im(M)/2}$\, where $C(M)$ is the same\nconstant as above.\nWe are going to 
 prove the same Weyl’s asymptotic estimates for $K_{\\infty}$-finite cusp
  forms for\nthe above space.\n
LOCATION:https://researchseminars.org/talk/UAANTS/32/
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