[Paper Review] Dirac cohomology for symplectic reflection algebras

Dan Ciubotaru|arXiv (Cornell University)|Feb 19, 2015

Advanced Algebra and Geometry17 references3 citations

TL;DR

This paper introduces a uniform framework for Dirac operators and Dirac cohomology in Drinfeld's Hecke algebras, generalizing earlier work on graded affine Hecke algebras. It applies this to symplectic reflection algebras, particularly rational Cherednik algebras, establishing a Dirac morphism that identifies the Calogero-Moser space's 0-fiber with the image of irreducible representations under a map to central characters, linking it to Lusztig-Rouquier families.

ABSTRACT

We define uniformly the notions of Dirac operators and Dirac cohomology in the framework of the Hecke algebras introduced by Drinfeld. We generalize in this way the Dirac cohomology theory for Lusztig's graded affine Hecke algebras. We apply these constructions to the case of symplectic reflection algebras defined by Etingof-Ginzburg, particularly to rational Cherednik algebras for real or complex reflection groups with parameters t,c. As applications, we give criteria for unitarity of modules in category O and we show that the 0-fiber of the Calogero-Moser space admits a description in terms of a certain "Dirac morphism" originally defined by Vogan for representations of real semisimple Lie groups.

Motivation & Objective

• To generalize Dirac cohomology theory from graded affine Hecke algebras to the broader class of Drinfeld's Hecke algebras.
• To establish a uniform construction of Dirac operators and Dirac cohomology in this generalized setting, requiring only a nondegenerate W-invariant symmetric bilinear form on the underlying vector space.
• To apply the theory to symplectic reflection algebras, especially rational Cherednik algebras for real and complex reflection groups.
• To provide criteria for unitarity of modules in category $\mathcal{O}$, refining previous non-unitarity results.
• To show that the 0-fiber of the generalized Calogero-Moser space arises as the image of a Dirac morphism, linking it to representation-theoretic partitions such as Lusztig-Rouquier families.

Proposed method

• Define a Dirac operator $\mathcal{D}$ on the tensor product of a Drinfeld Hecke algebra $\mathbf{H}_{\mathbf{a}}$ and a Clifford algebra $C(V)$, using a $W$-invariant symmetric bilinear form on $V$.
• Prove a formula for $\mathcal{D}^2$ that generalizes the classical Dirac operator square in representation theory.
• Establish an analogue of Vogan's conjecture for Dirac cohomology, showing that nonzero Dirac cohomology determines the infinitesimal character of a module.
• Construct a Dirac morphism $\zeta_c^*: \mathrm{Irr}(W) \to \Upsilon^{-1}(0)$, where $\Upsilon^{-1}(0)$ is the 0-fiber of the generalized Calogero-Moser space.
• Use the Dirac morphism to relate the partition of $\mathrm{Irr}(W)$ by central characters in $\mathbf{H}_{0,c}$ to the Lusztig-Rouquier families.
• Apply the theory to baby Verma modules $\bar{M}(\sigma)$ and show that the Dirac cohomology of $\bar{L}(\sigma)$ detects the image of $\sigma$ under $\zeta_c^*$.

Experimental results

Research questions

• RQ1 How can Dirac cohomology be uniformly defined for Drinfeld's Hecke algebras, extending the theory from graded affine Hecke algebras?
• RQ2 What is the structure of the Dirac operator square $\mathcal{D}^2$ in this generalized setting?
• RQ3 How does the Dirac morphism $\zeta_c^*$ relate the representation theory of $W$ to the geometry of the Calogero-Moser space?
• RQ4 Can the Dirac cohomology detect unitarity in category $\mathcal{O}$ for rational Cherednik algebras?
• RQ5 Does the Dirac morphism $\zeta_c^*$ identify the Calogero-Moser partition of $\mathrm{Irr}(W)$ with the Lusztig-Rouquier families?

Key findings

• The Dirac operator $\mathcal{D}$ satisfies a generalized square formula, $\mathcal{D}^2 = \text{Casimir-like term} + \text{central element}$, in the framework of Drinfeld's Hecke algebras.
• The analogue of Vogan's conjecture holds: nonzero Dirac cohomology determines the infinitesimal character of a module.
• The 0-fiber $\Upsilon^{-1}(0)$ of the generalized Calogero-Moser space is isomorphic to the image of the Dirac morphism $\zeta_c^*: \mathrm{Irr}(W) \to \Upsilon^{-1}(0)$, providing a geometric realization of the central character map.
• The morphism $\zeta_c^*$ is the $\varepsilon$-dual of the central character map $\Theta: \mathrm{Irr}(W) \to \Upsilon^{-1}(0)$, as shown in Corollary 5.10.
• For the Weyl group $B_2$ with constant parameter $c$, the representations $11\times 0$, $1\times 1$, and $0\times 2$ lie in the same fiber of $\Theta$, confirming they form a Lusztig family.
• The Dirac cohomology of the simple module $\bar{L}(\sigma)$ for $\sigma = 11\times 0$ in $B_2$ is $\sigma \otimes \bigwedge \mathfrak{h} = (11\times 0) + (1\times 1) + (0\times 2)$, confirming the Dirac morphism's nontrivial image.