Research statement. Pierre Clare

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1 Research statement Pierre Clare In 1896, Dedekind and Frobenius started representing elements of finite groups as matrices. The concept was soon extended to operators on general vector spaces and, in the 1940s, Gelfand used it as a means of introducing linear tools in the study of topological group actions by considering them at the level of function spaces: G X G L 2 (X) x g x f f(g 1 ) non-linear linear. Let G be a (topological) group. A unitary representation of G is a (strongly continuous) homomorphism π : G U(H) taking values in the unitary operators on a Hilbert space H. These objects are usually considered up to conjugation by unitary transformations, called intertwining or symmetrybreaking operators. The set of unitary equivalence classes of irreducible unitary representations is called the unitary dual of G. Its elements are the building blocks of harmonic analysis on G, a generalization of the theory of Fourier series on the circle, with far-reaching applications ranging from Number Theory to Physics. My research work concerns the application of methods of noncommutative geometry and geometric analysis in the representation theory of Lie groups. If G is locally compact, its unitary dual Ĝ is a prime example of noncommutative space, meaning that many of its properties can theoretically be recovered from the study of operator algebras naturally associated with G. In particular if G is a Lie group, the subset Ĝtemp Ĝ that supports the Plancherel measure is homeomorphic to the spectrum of the reduced C*-algebra of G: tempered dual reduced C*-algebra Ĝ temp C r(g). An ongoing project (joint with N. Higson) consists in studying Ĝtemp by analyzing C r(g). Our work has recently led to a new proof of the Connes-Kasparov isomorphism, a special case of the Baum-Connes conjecture, which describes the K-theory of C r(g) and can be thought of as a topological statement about Ĝtemp. Further applications of the C*-algebraic methods that we developed include: 1

2 (i) the analysis of the topological singularities of Ĝtemp; (ii) the construction of induced representations for infinite-dimensional Lie groups. For (ii), the process of induction consists in building representations of a group from representations of a subgroup. The problem of extending the Rieffel machinery to induce representations of infinite-dimensional gauge groups arises naturally in Physics in the context of quantization. My approach to this question involves host C*-algebras introduced by Grundling and Neeb, as well as amenable induction studied by Wolf and Ólafsson. The very definition of the unitary dual as a set of equivalence classes indicates that intertwining operators play a fundamental role in the theory. Unfortunately, even when they are known to exist, these operators are rarely given explicitly in functional models, which makes their use limited. Part of my work consists in using geometric and analytic methods to construct concrete symmetry-breaking operators between representations. This includes: (a) C*-normalization of Knapp-Stein standard integrals at the level of Hilbert modules (b) Kobayashi and Pevzner s F-method to determine differential symmetry-breaking operators on symmetric pairs (c) geometric Blattner pairings on co-adjoint orbits Because intertwining operators are ubiquitous in many ramifications of representation theory, these projects have numerous applications, in particular to aspects of the Langlands program (a), modular forms (b) and quantization (c). The overall objective of my research work is to develop further the understanding of unitary representations of groups from an analytic/geometric point of view. The results obtained so far indicate that this approach allows to streamline the presentation of known facts as well a tackle open problems in representation theory. Some of the projects presented presented below involve collaborations with several colleagues, in particular N. Higson (Penn State), G. Ólafsson (Louisiana State) and M. Pevzner (Reims). A more technical description of the problems presented below and how I am approaching them can be found in a recently submitted proposal for an NSF grant, available at math.dartmouth.edu/ clare/nsf16.pdf Note: throughout the text, numerical citations refer to publications I (co)authored, while alphabetical ones refer to other sources in the literature. 2

3 1 The tempered dual as a noncommutative space The research described in this section is part of an ongoing project, joint with N. Higson (Penn State) and T. Crisp (MPI Bonn) to use techniques of noncommutative geometry to study unitary representations of real reductive groups (typically, self-adjoint subgroups of GL(n, R)). Our goal is to understand the representation theory of such a group G in terms of the C*-algebras associated with it, such as the reduced C*-algebra C r(g), whose spectrum is the set Ĝtemp of (classes of) irreducible representations that occur in the decomposition of L 2 (G), which plays a central role in the harmonic analysis of G. A central tool in this investigation is the implementation of the functor of parabolic induction for group C*-algebras by means of Hilbert modules, introduced in [2] and based on ideas of M. Rieffel [Rie74]. It has allowed us to describe C r(g) in connection with the representation theory of G and led to a new proof of the Connes-Kasparov isomorphism [6, 7, 8]. Further objectives consist in studying adjoints for this functor by analogy with Bernstein s work in the p-adic case and eventually to reconstruct explicitly the Plancherel measure on the dual of G by operator-algebraic methods. Connes-Kasparov and the singularities of the tempered dual Harish-Chandra s celebrated philosophy of cusp forms gives a decomposition of the tempered dual Ĝtemp of a semisimple Lie group G into a disjoint union ( ) Ĝ temp = [P ] Ĝ P indexed by classes of parabolic subgroups P. Roughly speaking, Ĝ P consists of the irreducible components of representations parabolically induced from P, called principal series. The case P = G accounts for the discrete part of Ĝtemp (the discrete series) while the other terms encompass the continuous parts (generic principal series) as well as the singularities: the so-called limits of discrete series of G, responsible for the non-hausdorffness of Ĝtemp. One of the main results of [7] is a structure theorem for C r(g), that reflects Harish- Chandra s decomposition. We used the implementation of the functor of parabolic induction by means of the Hilbert modules that I introduced in [2] to establish an isomorphism ( ) C r(g) [P ] C r(g) P. 3

4 where the sum is indexed by the same subgroups that appear in ( ). Furthermore, we refined ( ) by decomposing each summand C r(g) P into a direct sum of fixed point algebras of compact operators on auxiliary Hilbert modules: ( ) C r(g) P σ K(E σ ) Wσ were σ runs over the discrete series representations of a subgroup of P and each W σ is a (finite) Weyl group whose action determines the equivalences among the representations parabolically induced from P. More recently, we were able to analyze further these algebras in order to calculate the K-theory of C r(g). More precisely, we proved in [8] that the C*-algebras ( ) are Morita equivalent to crossed-product of commutative algebras with subgroups of W σ. This allowed us to verify that the Connes-Kasparov map is an isomorphism along the lines suggested in A. Wassermann s short announcement [Was87]. The results discussed above rely on the reformulation of known facts from Representation Theory in the language of operators algebras. Because of these advances in the C*-algebraic description of Ĝtemp, we are now in a position to work in the reverse direction and use our machinery to obtain representation-theoretic results. An example of such results was already obtained by V. Lafforgue, who used the bijectivity of the Connes-Kasparov assembly map to give in [Laf02] a new and elegant proof of the classification by Harish-Chandra, the construction by Parthasarathy and the exhaustion by Atiyah and Schmid of the discrete series representations of connected semi-simple Lie groups. In particular, he obtained a bijection between the discrete series representations of G and a subset of the dual of K. Equipped with the refined analysis of C r(g) obtained in [7] and [8] it is natural to ask whether the singular points of Ĝtemp can be accounted for by a similar method, that is, to try and characterize the limits of discrete series by means of the Connes-Kasparov isomorphism. The main concept at play here is that of essential representation, which singles out the parameters σ in the decomposition ( ) which give non-zero contributions to the K-theory of C r(g). The work done in [8] has made the role of these representations in the C*- algebraic picture a lot clearer. The other ingredient we will use to tackle this problem is the dual-dirac element, similarly to the case studied in [Laf02]. 4

5 2 Induced representations for infinite-dimensional groups The main obstruction to apply the methods of the previous paragraph to the case of infinite-dimensional Lie groups is the fact that these groups are not locally compact hence do not automatically carry a Haar measure. For this reason, one cannot construct the maximal and reduced group C*-algebras C (G) and C r(g) obtained in the finite-dimensional case by completing convolution algebras. To remedy this situation, Grundling and Neeb have introduced the notion of host C*- algebra, defined for topological groups with no assumption of local compactness [GN14]. However, no general existence result for host algebras has been established so far and examples have been constructed in very few examples [NSZ15]. Motivation to extend the C*-algebraic framework to the representation theory of infinitedimensional Lie groups stems from constructions of principal series representations by J. Wolf [Wol05] for which unitarity is not known in general. Determining which of these representations can be obtained through C*-algebraic parabolic induction would be a significant progress on this problem. Moreover, the question of generalizing Rieffel induction to the infinite-dimensional appears naturally in Physics, more precisely in the case of gauge groups associated with certain particles [LW95]. Strategies to construct Rieffel modules in the absence of invariant measure include the use of invariant means. Concretely, this approach requires to study Hilbert module completions of certain spaces of bounded sections considered in [OW13, Wol13] where the concept of amenable induction is used in the case of direct limit groups. Another approach relies on the use of Gaussian pro-measures in the case of Hilbert-Lie groups. The results obtained in [LW95] suggest that quantized Poisson algebras may supply models of host algebras for certain classes of induced representations. Finally, in the case of loop groups, constructions of induction Hilbert modules may be obtained by studying imprimitivity of the representations hosted by the C*-algebras of [NSZ15]. This hybrid situation is of particular interest because the inducing representations come from finite-dimensional subgroups, to which the classical theory can be applied and yield ordinary Hilbert modules on which we can study the action of the ambient infinite-dimensional groups. Details about my approach to the problems outlined above can be found in in Section 2 of the recent grant proposal available at: math.dartmouth.edu/ clare/nsf16.pdf. Aspects of this particular project are studied jointly with G. Ólafsson (Louisiana State). 5

6 3 Symmetry-breaking operators Principal series representations play a crucial role in the representation theory of Lie groups. Therefore, a detailed understanding of intertwining operators between between these representations is critical, both from the classical and the C -algebraic point of view. Unfortunately, these intertwiners (Knapp-Stein operators) are given in ordinary functional models by singular integral kernels, which often makes their use limited in concrete situations. Over the past few years, new realizations of Knapp-Stein intertwiners have been obtained by means of classical geometric transforms see for instance [OP12]. Particularly simple expressions have been obtained in the so-called non-standard models of [KØP11] and [1], opening the way to a wide range of applications, including the study of branching laws [KØP11] and invariant trilinear forms [4]. The three research projects described below are intended to produce practical realizations of intertwining operators in various contexts: the C*-algebraic picture, by means of differential operators and the orbit method. C*-algebraic normalization of intertwining integrals Unitary equivalences among principal series representations appear in the C*-algebraic picture of the tempered dual of real reductive groups. In particular, the action of the Weyl group W σ in the decomposition ( ) of C r(g) is generated by families of classical Knapp- Stein intertwiners. However their action at the level of the Hilbert modules implementing parabolic induction has been described explicitly only in particular cases [3, 5]. The original work of Knapp and Stein [KS71, KS80] indicates that unitary intertwiners can be obtained by normalizing standard intertwining integrals of the form (1) I w f(g) = N f(gw n) d n where N is the unipotent radical of some parabolic subgroup, N = N and w is an element of the appropriate Weyl group. I proved in [2] that such integrals are defined on a dense subspace of functions in the parabolic induction Hilbert module E(G/N) but do not extend to Hilbert module operators. To circumvent this problem, I have introduced a notion of C*-algebraic normalization of these singular integrals [5]. By analogy with Knapp and Stein s original normalization procedure, one seeks twisted unitary operators U w : E(G/N) E(G/N) 6

7 such that the composition I w U w acts by a convolution operator Γ on a dense subset of E(G/N). Couples (U w, Γ) were explicitly obtained in special cases in [5, 3] and connections with the classical Knapp-Stein intertwiners were established. A remarkable feature of these constructions is the following: Observation. In all currently known cases, the normalized intertwiners U w can be defined by a Fourier integral operator on G/N. Similar observations have been made in the classical picture, for degenerate principal series representations induced from maximal parabolic subgroups in [1, 4] and [OP12], which suggest that an underlying principle is at work here. Further evidence comes from the case of p-adic groups, in which the construction of unitary intertwiners at the level of Hecke algebras does not present any analytic obstruction (see [BK02]). Moreover, a construction of Kazhdan [Kaz95] allows to produce unitary intertwiners for a large class of groups by reduction to the case of SL(2). A possible approach to construct Fourier-type unitary intertwiners on E(G/N) for general rank-one groups is to adapt Kazhdan s method. The main technical challenge in this direction will be to define direct integrals of Hilbert modules. Assuming that point resolved, Kazhdan s method is expected to extend with minor modifications. In higher rank, the situation is more delicate and additional ideas should be necessary, in view of the conjecture made at the end of [3] that the question of the C*-algebraic normalization for general parabolic subgroups of SL(n) is essentially equivalent to the generation of Rankin-Selberg γ-factors, a notoriously difficult problem connected with the representation-theoretic aspects of the Langlands program (see [BK00]). Further details, including toy models for the problem in the higher rank situation, can be found in Section 1.3 of the grant proposal I submitted recently. It is available at math.dartmouth.edu/ clare/nsf16.pdf. 7

8 F-method in higher split-rank As explained above, obtaining concrete symmetry-breaking operators is a generally arduous task. In recent work [KP16a] T. Kobayashi and M. Pevzner have studied general conditions under which certain representations associated with symmetric pairs can be intertwined by differential operators. Furthermore, they have given in [KP16b] explicit formulæ for such differential operators for the six complex geometries arising from symmetric pairs of split rank one. Their approach relies on the F-method, which reduces the problem to finding polynomial solutions of certain differential equations. Motivated by our work on intertwining operators for symplectic groups [1, 4] we intend to extend their method to the symmetric pair of rank n (Sp(n, R) Sp(n, R), Sp(n, R)), in which we conjecture that the solutions will be generated by a single polynomial in n variables. We will first study the case n = 2 in which calculations are already non-trivial but seem tractable. The operators obtained in [KP16b] by means of the F-method are related to Rankin- Cohen operators which play an important role in the study of classical modular forms. The case we plan to investigate is expected to yield an alternate approach to recent results on Siegel modular forms [IKO12]. This project is joint with M. Pevzner (U. of Reims) and T. Kobayashi (U. of Tokyo). Blattner pairings and Shapovalov duality A guiding principle in A. Kirillov s celebrated orbit method to study representations of Lie groups is a correspondence between representations of a group and the orbits under its co-adjoint action. In [Bla75], Blattner established in the case of SL(2, R) a relation between a pairing, due to Kostant, Sternberg and himself (BKS), between half-density-valued sections that are covariant constant along transverse polarizations of a co-adjoint orbit (picture on the left), and Knapp-Stein intertwiners. This observation relied on a concrete calculation of the BKS pairing that yielded kernels analogous to the standard integrals considered by Knapp and Stein. On the other hand, generalized Verma modules constitute building blocks in the representation theory of Lie algebras. Introduced as a tool to study irreducibility questions, Shapovalov forms provide a duality for Verma modules with far-reaching applications. 8

9 Since the irreducibility properties of principal series representations are entirely encoded by Knapp-Stein intertwiners, it makes sense to investigate the relations between these operators defined for group representations and particular Shapovalov forms defined on the corresponding Lie algebraic objects. More precisely, we expect that the analogy between Knapp-Stein intertwiners and the BKS pairing on the orbits associated with principal series can be explained in terms of the Shapovalov duality between the associated Verma modules. (The relation between generalized Verma modules and spherical principal series representations is described geometrically in [ČSS01].) The first step in this direction will be to elucidate the correspondence between the BKS pairing on the one-sheeted hyperboloid and the Shapovalov duality between the Verma modules associated with the spherical principal series of SL(2, R) by explicit calculations. An ulterior motive for investigating this question is the perspective of manufacturing analytic tools for the study of covariant star-products on co-adjoint orbits. In the algebraic approach promoted by Etingof, Schiffmann, Alekseev and others, based on Drinfeld s ideas, Shapovalov duality plays an important role in relation to the associativity equation of a given invariant deformation quantization (see [AL05]). We expect the analytic approach via Knapp-Stein intertwiners to offer an alternate and possibly simpler way to handle these questions in terms of associated covariant symbolic calculi on those orbits. This is a joint project with M. Pevzner (U. of Reims). 9

10 Publications and bibliography Publications [1] P. Clare. On the degenerate principal series of complex symplectic groups. J. Funct. Anal., 262(9): , [2] P. Clare. Hilbert modules associated to parabolically induced representations. J. Operator Theory, 69(2): , [3] P. Clare. C*-algebraic intertwiners for degenerate principal series of special linear groups. Chin. Ann. Math. Ser. B, 35(5): , [4] P. Clare. Invariant trilinear forms for spherical degenerate principal series of complex symplectic groups. Internat. J. Math., 26(13): , 16, [5] P. Clare. C -algebraic intertwiners for principal series: case of SL(2). J. Noncommut. Geom., 9(1):1 19, [6] P. Clare, T. Crisp, and N. Higson. Adjoint functors between categories of Hilbert C*-modules. J. Inst. Math. Jussieu, published online. [7] P. Clare, T. Crisp, and N. Higson. Parabolic induction and restriction via C*-algebras and Hilbert C*-modules. Compositio Math., 152: , [8] P. Clare and N. Higson. On the K-theory of the reduced C*-algebra of real reductive groups. In preparation, References [AL05] A. Alekseev and A. Lachowska. Invariant -products on coadjoint orbits and the Shapovalov pairing. Comment. Math. Helv., 80(4): , [BK00] A. Braverman and D. Kazhdan. γ-functions of representations and lifting, with an appendix by V. Vologodsky. In N. Alon, J. Bourgain, A. Connes, M. Gromov, and V. Milman, editors, Visions in Mathematics, pages Springer, Geom. Funct. Anal., Special Volume, Part I. [BK02] A. Braverman and D. Kazhdan. Normalized intertwining operators and nilpotent elements in the Langlands dual group. Moscow Mathematical Journal, 2(3): , [Bla75] [ČSS01] R. J. Blattner. Intertwining operators and the half-density pairing. In Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), pages Lecture Notes in Math., Vol Springer, Berlin, A. Čap, J. Slovák, and V. Souček. Bernstein-Gelfand-Gelfand sequences. Ann. of Math. (2), 154(1):97 113,

11 [GN14] [IKO12] [Kaz95] [KØP11] [KP16a] [KP16b] [KS71] [KS80] [Laf02] [LW95] [NSZ15] H. Grundling and K.-H. Neeb. Crossed products of C -algebras for singular actions. J. Funct. Anal., 266(8): , T. Ibukiyama, T. Kuzumaki, and H. Ochiai. Holonomic systems of Gegenbauer type polynomials of matrix arguments related with Siegel modular forms. J. Math. Soc. Japan, 64(1): , D. Kazhdan. Forms of the principal series for GL n. In Functional analysis on the eve of the 21st century, Vol. 1 (New Brunswick, NJ, 1993), volume 131 of Progr. Math., pages Birkhäuser Boston, Boston, MA, T. Kobayashi, B. Ørsted, and M. Pevzner. Geometric analysis on small unitary representations of GL(N, R). J. Funct. Anal., 260(6): , T. Kobayashi and M. Pevzner. Differential symmetry breaking operators: I. General theory and F-method. Selecta Math. (N.S.), 22(2): , T. Kobayashi and M. Pevzner. Differential symmetry breaking operators: II. Rankin- Cohen operators for symmetric pairs. Selecta Math. (N.S.), 22(2): , A. W. Knapp and E. M. Stein. Intertwining operators for semisimple groups. Ann. of Math., 93: , A. W. Knapp and E. M. Stein. Intertwining operators for semisimple groups II. Invent. Math., 60(1):9 84, V. Lafforgue. Banach KK-theory and the Baum-Connes conjecture. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages Higher Ed. Press, Beijing, N. P. Landsman and U. A. Wiedemann. Massless particles, electromagnetism, and Rieffel induction. Rev. Math. Phys., 7(6): , K.-H. Neeb, H. Salmasian, and C. Zellner. Smoothing operators and C -algebras for infinite dimensional lie groups. Preprint arxiv: , [OP12] [OW13] G. Ólafsson and A. Pasquale. The Cos λ and Sin λ transforms as intertwining operators between generalized principal series representations of SL(n + 1, K). Adv. Math., 229(1): , G. Ólafsson and J. A. Wolf. Separating vector bundle sections by invariant means. In Geometric analysis and integral geometry, volume 598 of Contemp. Math., pages Amer. Math. Soc., Providence, RI, [Rie74] M. A. Rieffel. Induced representations of C -algebras. Advances in Math., 13: , [Was87] A. Wassermann. Une démonstration de la conjecture de Connes-Kasparov pour les groupes de Lie linéaires connexes réductifs. C. R. Acad. Sci. Paris, 18(304): , [Wol05] J. A. Wolf. Principal series representations of direct limit groups. Compos. Math., 141(6): , [Wol13] J. A. Wolf. Principal series representations of infinite dimensional Lie groups, II: construction of induced representations. In Geometric analysis and integral geometry, volume 598 of Contemp. Math., pages Amer. Math. Soc., Providence, RI,