RIT: Representation Theory of Real Reductive Groups

Fall 2006

Organizers:
Jeffrey Adams
Thomas Haines

Meeting Time: Friday, 1:30-2::00
Room: MTH 1311


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:
  1. 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).
  2. 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).
  3. Chap. I: Sections 5-6: Statement and proof of Peter-Weyl theorem; Def. of induced representation. Statement only of Theorem 1.14 (Frobenius reciprocity).
  4. Chap. III: Sections 1-2: present all this material on universal enveloping algebras.
  5. Chap. III: Sections 3-4: C-infty-vectors and their density properties (incl. Cor.3.16).
  6. 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).
  7. 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.
  8. 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).
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. 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