Degenerate Principal Series for Symplectic and by Chris Jantzen PDF

By Chris Jantzen

This memoir experiences reducibility in a undeniable classification of precipitated representations for $Sp_{2n}(F)$ and $SO_{2n+1}(F)$, the place $F$ is $p$-adic. particularly, it truly is taken with representations bought via inducing a one-dimensional illustration from a maximal parabolic subgroup (i.e., degenerate vital sequence representations). utilizing the Jacquet module strategies of Tadic, the reducibility issues for such representations are decided. whilst reducible, the composition sequence is defined, giving Langlands information and Jacquet modules for the irreducible composition components.

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Lp L ( i / - t p , i / - t + l « ( p , 2), i/-t+f «(p, 2 ) , . . , i/"i«(p, 2); T ) DEGENERATE PRINCIPAL SERIES 27 (d) 2 < j < n~2 2 S(P)TTI = v-n+J+1p 0 L([i/-n+^'+2p, ^~V], K V , ^~V]; p x *) -fi/- J > 0 L([i/- n + J ' + 1 p, v~xp\, [v~j+lp, v~lp]',P * *) s(p)7r2 - i z - ^ ' + V 0 L([i/-"+' + 2 p, y'2p\ [v~jP, v-lp]\T) +p-ip 0 L([z/- n+ ''+V, i/" 2 p], [^~ j+1 p, v~xft,T) S{P)TT3 = i z - ^ ' + V ® L ( [ i / " ^ + 2 p , i/^'- 2 p], i/-'"-*«(p, 2 ) , . . , i/-ltf(p, 2); T ) + i / - ' p 0 L ( [ i / - n + ' + 1 p , ^ " V ] , ^ ~ J + ^ ( p , 2 ) , .

1}, the inductive hypothesis tells us that the three terms in S(p)7r are all irreducible. , is~a((p,2£) x v~z((p,£) ® a and vaC(p, 2£) x z/~2£(p, £) (gxj-are irreducible (cf. 1). The first of these contains terms of the form v~a~~£+2p . . and v~e+2p ® . . in its minimal Jacquet module, smin. £) a. Thus, all three terms in s^n come from the same component of n. Thus, n must be irreducible. If a is an integer, we have a < —2£. We can then use exactly the same argument as in the noninteger case-the i = 0,2£ terms from SQL^ are still irreducible and the argument goes through.

If p 0 9^ Po? 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).

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