Download e-book for iPad: Degenerate Principal Series for Symplectic Groups by Chris Jantzen

By Chris Jantzen

ISBN-10: 0821825496

ISBN-13: 9780821825495

This paper is worried with caused representations for $p$-adic teams. particularly, Jantzen examines the query of reducibility within the case the place the inducing subgroup is a maximal parabolic subgroup of $Sp_{2n}(F)$ and the inducing illustration is one-dimensional. various methods to this challenge are used. the 1st, in keeping with the paintings of Casselman and of Gustafson, reduces the matter to the corresponding query approximately an linked finite-dimensional illustration of a undeniable Hecke algebra. the second one technique relies on a means of Tadic and comprises an research of Jacquet modules. this can be used to procure a extra common end result on brought on representations, that could be used to house the matter whilst the inducing illustration satisfies a regularity situation. an identical simple argument can be utilized in a case-by-case style to nonregular instances.

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Matrix inside always lies in B (see appendix). C a s e 3 : w — wu For notational convenience, we set S = 7r(A^)^(wn).

W I 1 mod Vh -TW I I \ since m a x X > m a x W and the constraint Z € Vh h x lie in V ~ . , h Multiplying by Y then puts them in V . Thus, we get $(u ) = §(uxuzuYuw) (( ! = $ 1+ IZ tiytijy (1 + XZ)-1 = $ \l + XZ (1 + XZ)" Uy^W Chris Jantzen 26 since XZ G Vh~\ (1 + XZ)~XX = I m o d Vh. Thus, (i + xz)- 1 $( w -) = $ X 1 + XZ Y W V x-^i + x^^Cu-) The rest of the argument is similar to case A. As Z runs over Vh h x 1 runs over V ~ . Then, $ ( « -) = x " ^ ^ ^ ^ ) ^ ! ^ " ) f o r a11 s u c n l *, XZ ^ implies $(u") = 0, 1 1 since x is not constant on 1 + P' " .

Degenerate Principal construct a basis for VB*. Series 15 The problem breaks itself into three cases. 4. The reader is advised that the third case is substantially longer than the other two. We start with the following lemma, which is sort of a Bruhat decomposition. We let B be the parahoric subgroup corresponding to P (obtained by adjoining 5 2 , . . , sn to the Iwahori). 1 Let w00 = identity, I ( 1. u>io '1 •1 wn '1 \ 0 \ (wio = si). Note that for n = 1, u>10 does not exist. K = B U (Bw10B) U (BwuB) Then, for n > 2 and K = B U (BwnB) Proof for n = 1.

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Degenerate Principal Series for Symplectic Groups by Chris Jantzen


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