# Weil’s Representation and the Spectrum of the Metaplectic by Stephen S. Gelbart

By Stephen S. Gelbart

Booklet via Gelbart, Stephen S.

Similar linear books

Mengentheoretische Topologie

Eine verständliche und vollständige Einführung in die Mengentheoretische Topologie, die als Begleittext zu einer Vorlesung, aber auch zum Selbststudium für Studenten ab dem three. Semester bestens geeignet ist. Zahlreiche Aufgaben ermöglichen ein systematisches Erlernen des Stoffes, wobei Lösungshinweise bzw.

Combinatorial and Graph-Theoretical Problems in Linear Algebra

This IMA quantity in arithmetic and its purposes COMBINATORIAL AND GRAPH-THEORETICAL difficulties IN LINEAR ALGEBRA is predicated at the lawsuits of a workshop that used to be an essential component of the 1991-92 IMA application on "Applied Linear Algebra. " we're thankful to Richard Brualdi, George Cybenko, Alan George, Gene Golub, Mitchell Luskin, and Paul Van Dooren for making plans and enforcing the year-long application.

Linear Algebra and Matrix Theory

This revision of a widely known textual content comprises extra subtle mathematical fabric. a brand new part on purposes presents an advent to the trendy remedy of calculus of a number of variables, and the idea that of duality gets improved insurance. Notations were replaced to correspond to extra present utilization.

Extra resources for Weil’s Representation and the Spectrum of the Metaplectic Group

Example text

The group GL2(0v) [oi b1 ] (b is generated by the matrices e ~ W and i 0 (a [o a ] ~ Uv). 51) for these generators. Recall that our canonical additive character 0v. In particular, t e U v. ~v(tbq(X)) = i Therefore if rv(q)(l b)~~ the Fourier transform of i0 q(X) e O v, = ~v~176 is i0 . V Thus o o rv(q)([ 01 a])~v(X,t) = lal-i/2~~ obvious since a To define G~, b e 0v, and Note also that rv(q)(w)~~ ) = ~v~ ta -I) = oix ' t) is U v. rq globally, and to introduce theta-functions we need first to define an appropriate • ~.

U'(w) denote 31 Then U'(Wl)U'(w2) = F(Wl, W2)U'(Wl+W2) where (Wl, W 2) = if W i = (Ui, U[). That is, U' of G • G* with multiplier I/F U(w,t) comprises (w e G ~ G, t e T) law given by (wl, tl)(w2, t 2) = (Wl+W2, F(Wl,W2)tlt2). 34) is an irreducible is the unique irreducible or as the fixed. Therefore representation; representation of at least the first part of the result below is plausible. 20 automorphisms the normalizer of of (Segal). A(G) Let B0(G ) leaving ~ in T denote the group of pointwise L2(G), fixed, and let Bo(a) be the natural projection.

30). ) is a cusp form. 3) If(z)IIm(z) k/\$ < M~ Nhat we want to show is that such forms correspond to particularly nice functions on by ~ ~ The space of these forms will be denoted S(k/2, rl(N)). Recall that K denotes the maximal compact subgroup of whose connected component is K* = S0(2) = Jr(8) ~cos@ - sin@ = Lsin@ cos@ ]] GL2(~) 52 0 ~ ~ < 2~. with is still abelian. ~ Although To describe as ~/4~ ~ ~s above. K* with 0 ~ e ( 4w. K-~ must make elements ~* its character rather [~(e) = ~sine of pairs to [[r(e),~}] cose~ The isomorphism ~(0),-i} Z2 between correspond to these 7(2~) (and at most can have this property).