Metaplectic Groups and Segal Algebras by Hans Reiter

By Hans Reiter

Those notes provide an account of modern paintings in harmonic research facing the analytical foundations of A. Weil's concept of metaplectic teams. it truly is proven that Weil's major theorem holds for a category of services (a yes Segal algebra) greater than that of the Schwartz-Bruhat services thought of by way of Weil. the theory is derived right here from a few normal effects approximately this type which appears to be like a slightly typical one within the context of Weil's conception. No past wisdom of the latter is thought, besides the fact that, and the idea is built the following, step-by-step; additional, a whole dialogue of the Segal algebra involved is given, with references to the literature. Weil's metaplectic teams are a bit more straightforward to enquire whilst the attribute isn't 2; the case of attribute 2 provides a few specific beneficial properties that are absolutely mentioned. New difficulties that come up are indicated.

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Certain d i f f i c u l t i e s arise when V^ is r e s t r i c t e d but not basic and we w i l l use the followin g notation. For V^ r e s t r i c t e d w r i t e V i = V i s <3 V ^ , where V^ = V^ s unless X has type B n ; C n , F 4 ; or G2, w i t h p = 2 , 2 ; 2 ; 3 ; respectively. 6) ( so V^ s is basic and V^ is p-basic ). We w i l l sometimes w r i t e V i ~ to indicate one of V^; V^ s ; or V^1. For each r € Tt(Y)-Tt(Ly) we define a certain normal subgroup K y ^ of P Y ; which in most cases is just the largest normal subgroup of Py which is contained in Qy and does not contain the T y - r o o t corresponding to - r .

Then the weights of V are X a , X^-a, X a - a : - 2 £ ; \ - 2 a - 2 J 3 . If rf(L) = { a } , then we see there are no weights of Q-level 1. 4). ,Qk and let Q* = Q^ X ... X Qk. Then f o r d > 0, [Q* d , V i S L . ® [Q k d k, V k ] , the sum ranging over sets of nonnegative integers d ^ , . . +d k = d. Proof. It w i l l suffice to establish the result f o r k = 2. Clearly, 30 GARYM. SEITZ [ Q l d l , y±l ® [ Q 2 d 2 , V 2 ] < [ Q d l + d 2 , Vj_ ® V 2 ] . For just use d] elements f r o m Qj_, followed by d 2 elements f r o m Q 2 , and take commutators.

V k ] = (**) 2 ( v i ® . . ®V| < ) Mow (mod M). [ Q ; [ Q * d " 1 , V 0 ] ] = S is T - i n v a r i a n t , so S+M is a sum of T - w e i g h t spaces and we consider the sum of t e r m s in the r i g h t side of (**) corresponding to a fixed weight. ®V k ), where ] r qj = j s q^ for each pair of term s in the sum. and so j r ^ js. (***) For r ^ s, q* ^ qi Letting t vary we conclude each summand in ( * * * ) is in S+M. It f o l l o w s that r + M € S + M. This shows [ Q , ^ 1 ^ ] ] = [Q* d ,V 0 ] (mod M).

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