sing
.pdfLECTURES ON REPRESENTATIONS OF p-ADIC GROUPS |
21 |
where Wi is the set of all w such that w(i) < w(i + 1). This, however, is equivalent to
ww0(i) < ww0(j), which happens if and only if C+ and ww0(C+) are on the same side of the |
|
hyperplane fxi = xjg. Summarizing, we have |
M |
IndMG i (η)N = |
δ1/2χw. |
w(C+) {xi>xj}
¤
Corollary 13.3. Let χ be a regular character, and let Ω1, . . . Ωm be the connected components
of Ωχ. Then IndGB(χ) has m irreducible subquotients V1, . . . , Vm so that
M
(Vi)N = |
δ1/2χw. |
|
w(C+) Ωi |
Exercise. Show that IndG(δ1/2) has 2n |
subquotients. Hint: Try first the special case of |
B |
|
GL3(F ). |
|
References
[1]J. Bernstein and A. Zelevinsky, Representations of the group GLn(F), where F is a non-archimedean local field, Russian Math. Surveys 31 (1976), 1-68.
[2]J. Humphreys, Introduction to Lie Algebras and representation theory, Graduate Texts in Mathematics 9, Springer-Verlag, 1978.
[3]N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta Functions, Second Edition, Graduate Texts in Mathematics 58, Springer-Verlag, 1984.
[4]F. Rodier, D´ecomposition de la s´erie principale des groupes r´eductifs p-adiques, 408-424, Lecture Notes in Mathematics 880, Springer-Verlag, 1981.
[5]M. Tadi´c, Representations of classical p-adic groups, 129-204, Pitman Res. Notes Math. Ser. 311, Longman, Harlow 1994.
Department of Mathematics, University of Utah, Salt Lake City, UT 84112
E-mail address: savin@math.utah.edu