On the dimensions of spaces of Siegel modular forms of weight one
type: Item
instance of: scholarly article
title: On the dimensions of spaces of Siegel modular forms of weight one
zbMATH Open document ID: 0863.11033
DOI: 10.1007/BF02249262
author: Jian-Shu Li
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/58235
review text: Let \(G=Sp_n(\mathbb{R})\) be the real symplectic group of degree \(n\), and, given a positive integer \(m\), let \(\Gamma(m)\) be the principal congruence subgroup of \(Sp_n(\mathbb{Q})\) of level \(m\). For any integer \(k\geq 0\) let \(\pi_k\) be the irreducible holomorphic unitary representation of \(G\) having lowest weight \((k+1,1,\dots,1)\). Thus automorphic forms of type \(\pi_k\) correspond to classical Siegel modular forms with values in the \(k\)th symmetric power twisted by the determinant; in particular \(\pi_0\) corresponds to scalar valued Siegel modular forms of weight one. For \(k\geq n\), \(\pi_k\) is a unitary representation with non-vanishing relative Lie algebra cohomology, with twisted coefficients if \(k>n\). If \(\pi\) is an irreducible unitary representation of \(G\) let \(m(\pi,\Gamma(m))\) denote the multiplicity with which \(\pi\) occurs in the discrete spectrum of \(L^2(\Gamma(m) \setminus G)\). Suppose \(n\geq 3\). Then as shown \(m(\pi_k,\Gamma(m))=0\) unless either \(4\mid m\), or \(m\) has a prime factor \(p\) with \(p\equiv-1\bmod 4\). For an odd prime \(p>3\) with \(p\equiv-1\bmod 4\) and \(r\geq 1\) the following formula for the multiplicity \(m(\pi_k, \Gamma(p^r))\), \(k>0\), holds: \[ m(\pi_k, \Gamma(p^r))= (h/2)(p^{n(2r-1)}+ (-1)^k), \] where \(h\) denotes the class number of the field \(\mathbb{Q}(\sqrt{-p})\). A similar formula holds for \(k=0\). As a consequence, there is an analogous formula for the dimension of the \(L^2\)-cohomology \(H^n_{(2)} (\Gamma(p^r) \setminus H_n)\) of the associated locally symmetric space \(\Gamma(p^r) \setminus H_n\) [\(H_n\) the Siegel upper half space of degree \(n\)]. The method behind the proof is the following: The representations \(\pi_k\) come up in a list of all irreducible unitary representations of \(G\) which are of rank 2 in the sense of \textit{R. Howe} [On a notion of rank for unitary representations of classical groups, in: C.I.M.E. Summer School in Harmonic Analysis, 223-331 (1…
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 912462
zbMATH Keywords: Siegel modular forms of weight one
zbMATH Keywords: unitary representation
zbMATH Keywords: multiplicity
zbMATH Keywords: dimension of the \(L^ 2\)-cohomology
zbMATH Keywords: anisotropic binary forms
zbMATH Keywords: automorphic representations
reviewed by: Joachim Schwermer
MaRDI profile type: Publication
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OpenAlex ID: W1997720618
description: scientific article; zbMATH DE number 912462