Dirac Induction Research Articles

An equivariant orbifold index for proper actions

For a proper, cocompact action by a locally compact group of the form H×G, with H compact, we define an H×G-equivariant index of H-transversally elliptic operators, which takes values in KK∗(C∗H,C∗G). This simultaneously generalises the Baum–Connes analytic assembly map, Atiyah’s index of transversally elliptic operators, and Kawasaki’s orbifold index. This index also generalises the assembly map to elliptic operators on orbifolds. In the special case where the manifold in question is a real semisimple Lie group, G is a cocompact lattice and H is a maximal compact subgroup, we realise the Dirac induction map from the Connes–Kasparov conjecture as a Kasparov product and obtain an index theorem for Spin-Dirac operators on compact locally symmetric spaces.

Construction of discrete series representations of via algebraic Dirac induction

The notion of algebraic Dirac induction was introduced by P. Pandžić and D. Renard. This is a construction which gives representations with prescribed Dirac cohomology. They proved that all holomorphic discrete series representations can be constructed via Dirac induction. All discrete series representations of the group are nonholomorphic. In this paper we prove that they can also be constructed using algebraic Dirac induction.Communicated by Silvana Bazzoni

Orbital integrals and K-theory classes

Let $G$ be a semisimple Lie group with discrete series. We use maps $K_0(C^*_rG)\to \mathbb{C}$ defined by orbital integrals to recover group theoretic information about $G$, including information contained in $K$-theory classes not associated to the discrete series. An important tool is a fixed point formula for equivariant indices obtained by the authors in an earlier paper. Applications include a tool to distinguish classes in $K_0(C^*_rG)$, the (known) injectivity of Dirac induction, versions of Selberg's principle in $K$-theory and for matrix coefficients of the discrete series, a Tannaka-type duality, and a way to extract characters of representations from $K$-theory. Finally, we obtain a continuity property near the identity element of $G$ of families of maps $K_0(C^*_rG)\to \mathbb{C}$, parametrised by semisimple elements of $G$, defined by stable orbital integrals. This implies a continuity property for $L$-packets of discrete series characters, which in turn can be used to deduce a (well-known) expression for formal degrees of discrete series representations from Harish-Chandra's character formula.

Dirac Induction for Rational Cherednik Algebras

AbstractWe introduce the local and global indices of Dirac operators for the rational Cherednik algebra $\mathsf{H}_{t,c}(G,\mathfrak{h})$, where $G$ is a complex reflection group acting on a finite-dimensional vector space $\mathfrak{h}$. We investigate precise relations between the (local) Dirac index of a simple module in the category $\mathcal{O}$ of $\mathsf{H}_{t,c}(G,\mathfrak{h})$, the graded $G$-character of the module, the Euler–Poincaré pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for $\mathsf{H}_{t,c}(G,\mathfrak{h})$ constructed from finite-dimensional $G$-modules. We define and compute the index of a Dirac operator on the integral-reflection module and show that the index is, in a sense, independent of the parameter function $c$. The study of the kernel of these global Dirac operators leads naturally to a notion of dualised generalised Dunkl–Opdam operators.

An equivariant index for proper actions II: Properties and applications

In the first part of this series, we defined an equivariant index without assuming the group acting or the orbit space of the action to be compact. This allowed us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In this paper, we investigate properties and applications of this index. We prove that it has an induction property that can be used to deduce various other properties of the index. In the case of compact orbit spaces, the index is a special case of Kasparov’s index of transversally elliptic operators. In that case, we show how it is related to the analytic assembly map from the Baum–Connes conjecture, and an index used by Mathai and Zhang. In the case of noncompact orbit spaces, we use the index to define a notion of K -homological Dirac induction, and show that, under conditions, it satisfies the quantisation commutes with reduction principle.

Algebraic and analytic Dirac induction for graded affine Hecke algebras

AbstractWe define the algebraic Dirac induction map ${\mathrm{Ind} }_{D} $ for graded affine Hecke algebras. The map ${\mathrm{Ind} }_{D} $ is a Hecke algebra analog of the explicit realization of the Baum–Connes assembly map in the $K$-theory of the reduced ${C}^{\ast } $-algebra of a real reductive group using Dirac operators. The definition of ${\mathrm{Ind} }_{D} $ is uniform over the parameter space of the graded affine Hecke algebra. We show that the map ${\mathrm{Ind} }_{D} $ defines an isometric isomorphism from the space of elliptic characters of the Weyl group (relative to its reflection representation) to the space of elliptic characters of the graded affine Hecke algebra. We also study a related analytically defined global elliptic Dirac operator between unitary representations of the graded affine Hecke algebra which are realized in the spaces of sections of vector bundles associated to certain representations of the pin cover of the Weyl group. In this way we realize all irreducible discrete series modules of the Hecke algebra in the kernels (and indices) of such analytic Dirac operators. This can be viewed as a graded affine Hecke algebra analog of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and to Atiyah and Schmid.

Dirac cohomology and Dirac induction

Let G be a connected semisimple Lie group with a maximal compact group K of equal rank. We use the Dirac cohomology of the unitary representations to define Dirac-induction from a representation of K to the discrete series of G. This is closely related to the Dirac induction for the reduced group C*-algebras C* red(G) and a geometric construction of discrete series for semisimple Lie groups. Furthermore, we use Dirac cohomology of the Kostant’s cubic Dirac operator to define Dirac-induction for compact Lie groups. This induction for compact Lie groups is simpler than the Bott’s induction and is easier for calculation.

Dirac Induction for Loop Groups

Using a coset version of the cubic Dirac operators for affine Lie algebras, we give an algebraic construction of the Dirac induction homomorphism for loop group representations. With this, we prove a homogeneous generalization of the Weyl–Kac character formula and show compatibility with Dirac induction for compact Lie groups.

Twisted representation rings and Dirac induction

Extending ideas of twisted equivariant K-theory, we construct twisted versions of the representation rings for Lie superalgebras and Lie supergroups, built from projective Z 2 -graded representations with a given cocycle. We then investigate the pullback and pushforward maps on these representation rings (and their completions) associated to homomorphisms of Lie superalgebras and Lie supergroups. As an application, we consider the Lie supergroup Π ( T * G ) , obtained by taking the cotangent bundle of a compact Lie group and reversing the parity of its fibers. An inclusion H ↪ G induces a homomorphism from the twisted representation ring of Π ( T * H ) to the twisted representation ring of Π ( T * G ) , which pulls back via an algebraic version of the Thom isomorphism to give an additive homomorphism from K H ( pt ) to K G ( pt ) (possibly with twistings). We then show that this homomorphism is in fact Dirac induction, which takes an H-module U to the G-equivariant index of the Dirac operator / ∂ ⊗ U on the homogeneous space G / H with values in the homogeneous bundle induced by U.

Dirac Induction for Unimodular Lie Groups

In this paper, we give a geometric realization of discrete series representations for unimodular Lie groups on the spaces of harmonic spinors by using Connes–Moscovici'sL2-Index theorem. Our work is a continuation of Atiyah–Schmid's geometric realization of discrete series representations for semisimple Lie groups and Connes–Moscovici's realization of square-integrable representations for nilpotent Lie groups.