[Paper Review] Introduction to the Alexandru Conjecture

Pierre-Yves Gaillard|arXiv (Cornell University)|Mar 12, 2000

Finite Group Theory Research3 citations

TL;DR

This paper introduces the Alexandru Conjecture, a proposed analog of the Kazhdan-Lusztig theory for Harish-Chandra modules in the context of real reductive Lie groups. It extends results from the BGG category O—such as projective covers, Ext-polynomials, and Grothendieck group relations—to the setting of Harish-Chandra modules using a formal $ Z $-algebra structure, conjecturing analogous properties including polynomial invariants and categorically defined modules with endomorphism algebra $ \mathbb{C} $. The key contribution is a conjectural framework for computing Ext-algebras via new polynomials $ \widetilde{a}_{ij} $, generalizing Delorme and Kazhdan-Lusztig-Vogan polynomials.

ABSTRACT

Is a Verma module transformed into another Verma module by a selfequivalence? The answer is affirmative and the proof suggests a notion of standard object in the category of Harish-Chandra modules that coincides often, but not always, with the usual one.

Motivation & Objective

• To extend the structural theorems of BGG category $ \mathcal{O} $—such as projective covers and Ext-polynomials—to the setting of Harish-Chandra modules for real reductive Lie groups.
• To formulate a conjectural framework for the representation theory of Harish-Chandra modules using a formal $ Z $-algebra $ A $, where $ Z = \mathbb{C}[[z_1,\dots,z_r]] $, and $ A $-modules of finite dimension.
• To generalize the Grothendieck group relations and Ext-polynomial identities from category $ \mathcal{O} $ to the Harish-Chandra setting, particularly through the use of Delorme-type series and polynomial invariants.
• To propose a refined structure for the Ext-algebra of simple modules, conjecturing that it decomposes as a sum over $ \widetilde{d}_k \widetilde{a}_{ki} \widetilde{a}_{kj} $, with $ \widetilde{a}_{ij} $ defined via Langlands parameters and Kazhdan-Lusztig-Vogan polynomials.

Proposed method

• Define $ A $ as a complete, finitely generated $ Z $-algebra with $ \mathcal{H}_\rho \simeq A\text{-df} $, where $ \mathcal{H}_\rho $ is the category of Harish-Chandra modules with trivial infinitesimal character.
• Construct projective covers $ P_i $ of simple $ A $-modules $ L_i $, and define $ M_i = A e_i / \sum_{j > i} A e_j A e_i $, mimicking the Verma module construction in category $ \mathcal{O} $.
• Introduce the $ \widetilde{a}_{ij}(t) = t^{\widetilde{\ell}(j) - \widetilde{\ell}(i)} \widetilde{p}_{ij}(t^{-2}) $, where $ \widetilde{p}_{ij} $ are Kazhdan-Lusztig-Vogan polynomials and $ \widetilde{\ell}(i) $ is the complex dimension of the $ K_{\mathbb{C}} $-orbit associated to $ i $.
• Define $ \widetilde{d}_i = (1 - t^2)^{\dim \mathfrak{a}_i} $, a correction factor depending on the Langlands decomposition of the parabolic subalgebra $ \mathfrak{p}_i $.
• Conjecture that the Poincaré series of $ \mathop{\rm Ext}^\bullet(L_i, L_j) $ equals $ \sum_k \widetilde{d}_k \widetilde{a}_{ki} \widetilde{a}_{kj} $, generalizing the category $ \mathcal{O} $ result.
• Use the order $ \leq $ on index set $ I $ defined by $ i \leq j $ if $ \ell(j) = \ell(i) + 1 $ and $ \mathop{\rm Ext}^1(L_j, L_i) \neq 0 $, to define the poset structure for the conjectural theory.

Experimental results

Research questions

• RQ1 Can the structural theorems of BGG category $ \mathcal{O} $—such as the projective cover description and Ext-polynomial identities—be extended to the setting of Harish-Chandra modules for real reductive groups?
• RQ2 Is there a canonical $ Z $-algebra $ A $, complete and of finite type over $ Z = \mathbb{C}[[z_1,\dots,z_r]] $, such that $ \mathcal{H}_\rho \simeq A\text{-df} $, with $ A $ commutative modulo its radical?
• RQ3 Do the Grothendieck group relations in category $ \mathcal{O} $, such as $ \boldsymbol{L}_y = \sum_x a_{x,y}(-1) \boldsymbol{M}_x $, have a valid analog in the Harish-Chandra setting via $ \overline{M}_i $ and $ a_{ij} $?
• RQ4 Can the Ext-algebra of simple Harish-Chandra modules be expressed as a sum over $ \widetilde{d}_k \widetilde{a}_{ki} \widetilde{a}_{kj} $, with $ \widetilde{a}_{ij} $ defined via Langlands parameters and Kazhdan-Lusztig-Vogan polynomials?
• RQ5 Does the endomorphism algebra of the reduced projective cover $ \overline{M}_i $ equal $ \mathbb{C} $, as in the category $ \mathcal{O} $ case?

Key findings

• The conjectural module $ M_i = A e_i / \sum_{j > i} A e_j A e_i $ satisfies $ \mathop{\rm End}_A(M_i) = \mathbb{C} $, generalizing Theorem 2 from category $ \mathcal{O} $.
• The Grothendieck group relation $ \boldsymbol{L}_j = \sum_i a_{ij}(-1) \overline{\boldsymbol{M}}_i $ is conjectured to hold, where $ a_{ij} $ is the Poincaré series of $ \mathop{\rm Ext}_A^\bullet(M_i, L_j) $.
• The polynomials $ p_{ij} $ in the conjecture satisfy $ \deg p_{ij} < (\ell(j) - \ell(i))/2 $ for $ i < j $, and $ p_{ii} = 1 $, mirroring the properties of Kazhdan-Lusztig polynomials.
• The Ext-algebra of simple modules satisfies $ SP\, \mathop{\rm Ext}_A^\bullet(L_i, L_j) = \sum_k a_{ki} a_{kj} $, a direct analog of Theorem 5 in category $ \mathcal{O} $.
• Under the assumption that $ G $ and $ K $ have the same rank, the conjecture $ SP\, \mathop{\rm Ext}_A^\bullet(L_i, L_j) = \sum_k \widetilde{d}_k \widetilde{a}_{ki} \widetilde{a}_{kj} $ is proposed, with $ \widetilde{d}_i = (1 - t^2)^{\dim \mathfrak{a}_i} $ and $ \widetilde{a}_{ij} = t^{\widetilde{\ell}(j) - \widetilde{\ell}(i)} \widetilde{p}_{ij}(t^{-2}) $.
• The construction of $ \overline{M}_i $ as $ M_i / \text{rad}(\mathop{\rm End}_A M_i) \cdot M_i $ is introduced, and it is conjectured that $ \mathop{\rm End}_A(\overline{M}_i) = \mathbb{C} $, generalizing Theorem 2.