New PDF release: Degenerate Principal Series for Symplectic and

By Chris Jantzen

ISBN-10: 0821804820

ISBN-13: 9780821804827

This memoir experiences reducibility in a definite classification of brought about representations for $Sp_{2n}(F)$ and $SO_{2n+1}(F)$, the place $F$ is $p$-adic. particularly, it truly is enthusiastic about representations bought via inducing a one-dimensional illustration from a maximal parabolic subgroup (i.e., degenerate relevant sequence representations). utilizing the Jacquet module suggestions of Tadic, the reducibility issues for such representations are made up our minds. whilst reducible, the composition sequence is defined, giving Langlands info and Jacquet modules for the irreducible composition components.

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V (lpo,k) x a is irreducible (cf. 12). ). 5. C O M P O N E N T S : T H E " R A M I F I E D " CASE In this section, we study TT = va((po, k) x £(p, £\ a) for po ^ p. &) * cr. ) We are particularly interested in the cases where TT is a degenerate principal series representation. Let \ = | • l ^ o , a G R, DEGENERATE PRINCIPAL SERIES 39 be a one-dimensional representation of Fx. 12. If i/>0 = 1, the components of x o detk x tri (when reducible) are covered by the next section. Suppose ^o = sgn (order two).

0 if A: = 2£}}. Let 5i denote the first set; S2 the second. Suppose TT is reducible. 9, without loss of generality, we may restrict our attention to a < 0. 1. (1) aeSua#S2 7T = 7Ti + 7T2 w i t h 7Ti = L ( [ l / * + ~ ^ ~ p , Va+~2~p], [v~i+2p, V~*p\, a) TT2 = L([i/*+=lPp, i / ^ - t p ] , i/-'«(p, 2), i/"< +1 «(p, 2 ) , . . , i/*+i«(p, 2), [z/ a + ^p,zy-^p];cr) (2) a € S 2 , « 0 Si Writea = ~|+jf, 0 < j < | . (a) j = k-£(j

Further, by inductive hypothesis, we know that T', r", r'" decompose according to the theorem. The proof of the theorem is broken into subcases based on how r', T"', r'" decompose (with respect to the theorem). The particular case of the theorem governing the decomposition of r' is given in the second column in the table below, and is easily determined from k'—f, / , £'. One note: if j = | , in order to avoid having a' > 0, we replace r' = v~*~ p®vi({p, k — 1) x£(p, £\ cr) f with v^^p T '" I / - 5 C ( P , Jfc - 1) x C(P> 4 *) = T " (so then k' - / = *" - j " , f = j " , £' = £").

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Degenerate Principal Series for Symplectic and Odd-Orthogonal Groups by Chris Jantzen


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