References: Representation Theory of Semisimple Groups: An Overview Based
on Examples by Anthony
Knapp
Unitary Representations of Reductive Lie Groups by David Vogan
Lie Groups Beyond an Introduction by Anthony KNapp
Plan:
We are going to work up to the definition of admissible
representations in Chapter 8 of the Overview book.
Here is a detailed list of lectures:
Chap. I Sections 1-3: All material from Sections 1,2
(but skip Prop. 1.4). Sections 3 up through Schur's lemma
(Prop. 1.5).
Chap. I : Sections 3 after Schur's lemma; all of
Sections 4; Sections 5 to just before Peter-Weyl theorem (Thm
1.12). Proof Theorem 1.15 (state and use just the part of Peter-Weyl
theorem you need).
Chap. I: Sections 5-6: Statement and proof of
Peter-Weyl theorem; Def. of induced representation. Statement only of
Theorem 1.14 (Frobenius reciprocity).
Chap. III: Sections 1-2: present all this material on
universal enveloping algebras.
Chap. III: Sections 3-4: C-infty-vectors and their
density properties (incl. Cor.3.16).
Chap. IV: Sections 2-5: Roots, weights, and root
systems. Prop. 4.1 at least (a-c) stated. Definitions of Cartan
matrix, Dynkin diagram, and Weyl group. Examples (see Sections 1, and
p. 71,77).
Chap. IV: Sections 5,7: Prop. 4.13, Thm. 4.28 (Theorem
of highest weight). Definition of Verma module, and statement of
Prop. 4.34.
Chap V: Sections 1,2: Compact forms and the unitary
trick: in particular Prop. 5.3 and Prop. 5.7. Statement of Iwasawa
decomposition (Thm. 5.12).
Chap. V: Sections 5, Parabolic subgroups
(esp. Prop. 5.23) and Langlands decomposition. Chap. VII: Sections
1-2 three pictures for induced representations and their elementary
properties. If time: Chap VIII Sections 1 on motivation for
admissible representations.
Chap. VIII: Sections 2: Definition of admissible
representation and examples; in particular ``unitary irred implies
admissible'' (Thm 8.1); ``induced of unitary admissible are
admissible'' (Prop. 8.4); Prop. 85 showing that K-finite vectors are
Lie(G)-stable.
Chap. VIII: Sections 3: Casimir operator, and
infinitesimal equivalence. Theorem 8.9, which together its
corollaries 8.10 and 8.11 and earlier material shows that admissible
representations are ``the same'' as (Lie(G), K)-modules.
Chap. VIII: Sections 6: Infinitesimal characters,
esp. Prop. 8.21-8.22. Chap. IX Sections 1 up through Cor. 9.2, which
states that any 2 irreducible unitary representations that are
infinitesimally equivalent, and unitarily equivalent.
Faculty lecture: Langland's classification of
admissible representations. (?)
Schedule (tenative)
Lecture 1, September 8: Jeffrey Adams
Lecture 2, September 15: Walter Randolph Ray-Dulany
Lecture 3, September 22: Beth McLaughlin
Lecture 4, September 29: Chris Zorn
Lecture 5, October 6: Sean Rostami
Lecture 6, October 13: Moshe Adrian
Lecture 7, October 20: David Aulicino
Lecture 8, October 27: Ben Trahan
Lecture 9, November 3: Kevin Wilson
Lecture 10, November 10: Ted Clifford
Lecture 11, November 17: John Habert
Lecture 12, December 1: Moshe Adrian
Lecture 12, December 8: Sean Rostani
Lecture 13, December 11 (Monday): Conclusion
Here is a mailing list for the RIT:
jda@math, tjh@math, aulicino@math, madrian@math, ecliff@math, btrahan@math, bethmcl@math, kmwilson@math, raydlany@math, czorn@math, brenton@cs.wisc.edu, srostami@math, jhabert@math