Throughout the lecture, the basic reference is [PM05].
Reference:
[PM05] | T. Kobayashi, Restrictions of unitary representations of real reductive groups, Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces (J.-P. Anker and B. Ørsted, eds.), Progress in Mathematics 229, Birkhäuser, 2005, pp. 139-207. |
Section 0. Introduction — decompositions of representations
We explain our internal and external motivations of "decompositions of representations" in the line of [AMS05, S.1].
Section 1. Real reductive groups
Basic properties of real reductive Lie groups are summarized. These results are standard, and can be found in various standard textbooks such as [Knp02] and [Wal88]. Our exposition follows [PM05, Chapter 1].References:For the natural inclusive relations among classical groups, Dr. Alldrige will give a "complementary lecture" on reductive symmetric pairs on 19 July. We proposed that representation should be understood not only as those of a single group but also through its natural subgroups (e.g. reductive symmetric pairs).
[Knp02] | Anthony W. Knapp, Lie groups beyond an introduction, second edition, Progress in Mathematics 140, Birkhäuser, 2002. |
[PM05] | T. Kobayashi, Restrictions of unitary representations of real reductive groups, Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces (J.-P. Anker and B. Ørsted, eds.), Progress in Mathematics 229, Birkhäuser, 2005, pp. 139-207. |
[AMS05] | T. Kobayashi, Theory of discrete decomposable branching laws of unitary representations of semisimple Lie groups and some applications, Sugaku Expositions 18 (2005), 1-37. |
[Wal88] | Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics 132, Academic Press, 1988. |
Section 2. Decomposition of representations
2.1 | Continuous represntations (see the textbook [Wal88] for general results). |
2.2-2.4 | Direct integral of Hilbert spaces and unitary representations on it. For more details, we refer to the textbooks [GV4], [Y], and [Wal88]. |
[GV4] | I. M. Gel'fand and N. Ya. Vilenkin, Generalized functions, Vol. 4. Applications of harmonic analysis, Translated from Russian by Amiel Feinstein, Academic Press, 1964. |
[Wal88] | Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics 132, Academic Press, 1988. |
[Y] | Kosaku Yosida, Functional analysis, Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, 1995. |
2.4 | Irreducible decomposition (theorem of Mautner-Teleman) Example of L^{2}(R) |
2.5 | Admissible representations |
The concept of "admissible representations" in this lecture is more general than usual, and was introduced in [Inv94]. Admissible representations in the usual sense correspond to "K-admissible representations" in our terminology.References:Two important results: Gelfand-Piateski-Shapiro's theorem on the admissibility of L^{2}(G/Γ) and Harish-Chandra's admissibility theorem. The latter theorem became a foundation, from which the algebraic theory of (g,K)-modules has been developed. We gave a sketch of the microlocal proof of this theorem in place of the well-known algebraic proof using Jacquet functors.
See [PM05, Chapter 2] and references thereis.
[Inv94] | T. Kobayashi, Discrete decomposability of the restriction of A_{q}(λ) with respect to reductive subgroups and its applications, Invent. Math. 117 (1994), 181-205. |
[PM05] | T. Kobayashi, Restrictions of unitary representations of real reductive groups, Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces (J.-P. Anker and B. Ørsted, eds.), Progress in Mathematics 229, Birkhäuser, 2005, pp. 139-207. |
Section 3. SL(2,R) and branching laws
We illustrate unitary principal series representations π_{λ} of reductive Lie groups by the example of G=SL(2,R) and explore its branching laws when restricted to one dimensional subgroups N, A, and K. The combination of these branching laws leads us naturally to an analytic proof of the fact that π_{λ} are irreducible for all λ such that Re λ = −1 (unitary axis). This is a part of Bargman's classification (1947) of the unitary dual of SL(2,R). Our exposition follows [PM05, Chapter 3]. The general definition of (unitary) principal series representations is explained in the complementary lecture of Professor Hilgert on 19 July (Wed). Furthermore, the decomposition of the tensor product representation of two such principal series representations (this is another branching law concerning SL(2,R)) is explained in the complementary lecture of Professor Pasquale on 21 July (Fri).Reference:
[PM05] | T. Kobayashi, Restrictions of unitary representations of real reductive groups, Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces (J.-P. Anker and B. Ørsted, eds.), Progress in Mathematics 229, Birkhäuser, 2005, pp. 139-207. |
Section 4. Admissible restriction — Criterion
We gave a criterion for the restriction of irreducible unitary representations to reductive subgroups to be admissible, namely, the decomposition is discretely decomposable and mulitiplicities are finite.The original source is [Ann98]. See the exposition [PM05, Chapter 6]. Our 5th lecture consists of the following contents:
4.1 | Distribution characters and hyperfunction characters |
4.2 | Hyperfunction — two approaches (a) "boundary value" and (b) "analytic functionals". In the lecture, we explained the naive idea of (a). |
4.3 | Highest weight theory for groups. |
4.4 | Momentum image of the cotangent space of compact Lie groups. |
4.5 | Asymptotic cones. |
4.6 | Asymptotic K-support. |
4.7 | Criterion for the admissibility of the restriction (main result). |
References:
[Ann98] | T. Kobayashi, Discrete decomposability of the restriction of A_{q}(λ) with respect to reductive subgroups II — micro-local analysis and asymptotic K-support, Annals of Math. 147 (1998), no. 3, 709-729. |
[PM05] | T. Kobayashi, Restrictions of unitary representations of real reductive groups, Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces (J.-P. Anker and B. Ørsted, eds.), Progress in Mathematics 229, Birkhäuser, 2005, pp. 139-207. |
The 6th lecture explains an idea of the proof of the criterion for the admissibility. Our proof is based on [Ann98].References:The criterion comes from the transversal condition of the singularity spectrum of the hyperfunction character (a notion analogous to the wave front set for distributions) and the conormal bundle. This transversal condition justifies the restriction to be well-defined.
To give an upper estimate of the singularity spectrum, we investigate the complex domain of a complexified group to which the hyperfunction character extends holomorphically. The proof of this estimate uses the fact that hyperfunction has an infra-exponential growth in the Paley-Wiener theorem, which was explained (in the dual picture, i.e. hyperfunctions on compact manifolds are dual to analytic functions) in the complementary lecture by Professor Hilgert on July 17. Further, Professor S. Hansen and Dr. Troels Johansen may give complementary lessons on microlocal analysis in the second week.
Algebraic approach to discretely decomposable modules by means of (g,K)-modules is not discussed in this lecture. We refer to [Inv98] for the interested reader.
[Ann98] | T. Kobayashi, Discrete decomposability of the restriction of A_{q}(λ) with respect to reductive subgroups II — micro-local analysis and asymptotic K-support, Annals of Math. 147 (1998), no. 3, 709-729. |
[Inv98] | T. Kobayashi, Discrete decomposability of the restriction of A_{q}(λ) with respect to reductive subgroups III — restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1998), 229-256. |
Section 5. Geometric quantization of elliptic orbits
Original references are papers by Schmid (solution to the Kostant-Langland conjecture on the geometric construction of discrete series representations), Vogan's Green book (1981) on the construction of Zuckerman's derived functor modules, papers by Vogan and Wallach on the unitarization, and a paper by H. Wong on the closed range problem of the d-bar operator. Some of these results are purely algebraic.Reference:Our exposition follows [AMS98, Section 2] where we formalized their results in the spirit of the Kirillov-Kostant orbit method and the geometric quantization of coadoint orbits.
[AMS98] | T. Kobayashi, Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory, Translations, Series II, Selected Papers on Harmonic Analysis, Groups, and Invariants (K. Nomizu, ed.), vol. 183, Amer. Math. Soc., 1998, pp. 1-31. |
We give an application of the criterion of admissible restrictions to the topological problem of subvarieties (modular varieties) in locally symmetric spaces. For the original source together with its motivations, see [Oda98]. See also [AMS05, Section4] for the naive idea of the proof.Section 7. Applications to representation theoryThe Clifford-Klein form is a generalization of a closed Riemann surface of genus > 2, which we can regard as the double coset Γ\G/K of the triple Γ ⊂ G ⊃ K. Here, Γ is a cocompact torsion free discrete subgroup of G. After giving a brief summary of results by Gelfand-Piateski-Shapiro, Matsuchima-Murakami, Borel-Wallach, and Vogan-Zuckerman on the topology of the Clifford-Klein forms Γ\G/K, we consider the topology of the cycles defined by subgroups G' of G. An computation for (G, G') = (SO(4,2), SO(4,1)) is illustrated.
Since the general theory [Inv94] on admissible restrictions to non-compact subgroups, applications of admissible restrictions to internal problems of representations have also developed in various directions, ranging from the study of representations of larger groups and smaller groups (subgroups) to the branching laws of their own by Gross-Wallach, Lee-Loke, Duflo-Vargas, Speh, and myself.References:Some key words in the lecture are Stone Age (construct new tools by decomposing "stones"), Egg or Stone ? See by decomposing it!
See [AMS05, Section 4] and [ICM02] for details.
[Inv94] | T. Kobayashi, Discrete decomposability of the restriction of A_{q}(λ) with respect to reductive subgroups and its applications, Invent. Math. 117 (1994), 181-205. |
[Oda98] | T. Kobayashi and T. Oda, A vanishing theorem for modular symbols on locally symmetric spaces, Comment. Math. Helv. 73 (1998), 45-70. |
[ICM02] | T. Kobayashi, Branching problems of unitary representations, Proc. of ICM 2002, Beijing, vol. 2, 2002, pp. 615-627. |
[AMS05] | T. Kobayashi, Theory of discrete decomposable branching laws of unitary representations of semisimple Lie groups and some applications, Sugaku Expositions 18 (2005), 1-37. |
© Toshiyuki Kobayashi