Weil’s Representation and the Spectrum of the Metaplectic by Stephen S. Gelbart
By Stephen S. Gelbart
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Extra resources for Weil’s Representation and the Spectrum of the Metaplectic Group
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).