Prenotes

1. (Casselman) g, U(g), Z(g)

• definition of \g = \sl(2)
• definition of U(\g)
• PBW
• \g-represenations = U(\g) modules
• Casimir
• Z(\g) = S(\h)^W = \C[casimir].  (Exercise: \sl(n)).

2. (Trapa)  Finite-dimensional representations

• Passage from G-representataion to \g-representations
• Explicit treatment of f/d representations (coordinate with Milicic's Monday afternoon lecture)
• Characters
• Exercise: Sym(C^2)

3. (Trapa) Measure and Integration

• Haar measure, modular functions (general groups)
• KAN, KAK, KP
• the modular function for SL(2,R)
• Weyl integration formula for SL(2,R)

4. (Milicic) ODE's and regular singularities

• Dragan's notes (pdf)
  

5. (Milicic) L2 of the circle

6. (Casselman)  Geometry of upper half-plane and the disc

• Cayley Transform
• Metric, Laplacian, etc.  (tie in with first AF lecture on Monday)
  

Week #1

	9:30	11:00	Lunch	2:00	3:30
Monday	Geometry of SL(2) (C) [no representation theory] Iwasawa, Cartan, KAK, conjugacy classes, Haar measure, upper half plane, Weyl integration formula	Motivation I: Heuristic overview from point of view of finite groups (T) heuristics from finite group representations and Fourier analysis		Motivation II: Details of a single representation of SL(2,R) (C) Delta = 0, boundary values of functions of the disc, etc.	G-representations  to (\g,K)-modules (M) notions of irreducibility, need to consider nonunitary representations, irreducible implies multiplicity free as a K representation, smooth and analytic vectors (sketchy), definition of category of HC modules, relation to unitarity (sketchy)
Tuesday	Algebraic classification of irreducible HC mods (T)	Enveloping algebra, its center (T) Dixmier's Schur's Lemma, HC homomorphism, Osborne's Lemma, revisit definition of HC modules, classification.		Matrix coefficients: differential equations I (M), power series expansion, regular singularities, (A) dependence only on HC-modules, existence; (B) character of a HC-module (need smooth compactly supported K-biinvariant functions are dense in smooth functions on G); immediately get that the character is invariant; observe that characters determine irreducible representations and virtual representations (factors to K-group).	Matrix coefficients: differential equations II (M)
Wednesday	Principal series (C); definitions, computation of HC modules, reducibility, relation to algebraic classification of first day,  remark  on subrepresentation theorem, characters (need Weyl integration formula on G).	Intertwining operators (C), explicit formula (observe meromorphic dependence), limit formula for c-function		\n-homology, embeddings into ps, asymptotics (M) Frobenius reciprocity, leading exponents, uniqueness of Langlands representation, classification revisited.	General character theory I (T), characters of finite-dimensional representations (need Weyl integration formula), introduce distribution characters for infinite-deimsnional representations, compute characters of PS interms and express in terms of orbital integrals, conclude with discussion of characters as eigendistributions.
Thursday	Proof of regulatity theorem for characters following Atiyah-Schmid (M or T) analytic preliminaries a la A-S, reduction of Casimir in elliptic and hyperbolic situation, sketch that the denominator is locally L1, deduce from eigendistribution interpretation that the character differs from a locally L1 function by something  supported on the nilpotent cone, write down Casimir in nbhd of principal nilpotent and apply A-S preliminaries to show that the difference is supported at 0, get matching conditions, finish by analyzing distributions supported at a point.	Osborne Conjecture (T) need tensoring, coherent families and characters of PS here.		Characters of DS (M or T or C) on elliptic set compare with K-character, on hyperbolic set use Osborne's conjecture	Unitarity (T) Bargman's classification and modern viewpoint.
Friday	D-modules I (M), Borel-Weil-Bott for SL(2), TDOs on projective space, Beilinson-Bernstein equivalence.	D-modules II (M), Harish-Chandra sheaves, geometric classification, cohomological induction and duality		AF Motivation (C)	Free

Week #2

	Monday	Tuesday	Wednesday	Thursday	Friday
9:30	Whittaker models (T)	APW II (C)	L-groups, Vogan's L (?)	Other groups	Other groups
11:00	Arthur's Paley-Weiner (C)	metaplectic SL(2) (?)	Weil groups, L-parameters (?)	Other groups	Other groups
2:00	Student sessions (all week long)
3:30	Student sessions (all week long)

Week #3 (Casselman: email me your notes.)

	Monday	Tuesday	Wednesday	Thursday	Friday
9:30
11:00
2:00
3:30

The authors generously acknowledge the support of Booker's Foundation.