Let $\mathcal{O}_0$ be the principal block of the BGG category $\mathcal{O}$ for a finite dimensional simple Lie algebra over $\mathbb{C}$. For an element $w$ in the Weyl group $W$, let $\Delta_w$ denote the Verma module with highest weight $w_0w^{-1}\cdot 0$, where $w_0\in W$ is the longest element, and $\cdot$' denotes the dot-action.

It is well known that

$\mathrm{dim}(\mathrm{Hom}(\Delta_v, \Delta_w)) \leq 1$

This fact is pretty straightforward to prove algebraically. However, I do not know how to see this topologically. Namely, I do not know how to prove this via the interpretation of Verma modules as perverse sheaves on the flag variety.

I would be grateful if someone could explain how to see this fact topologically.

Added later: In response to Jim Humphreys comment let me add some motivation:

  1. In this regard I think of category $\mathcal{O}$ as a "toy example". I would like to know what sort of generality this fact holds for. For instance, is the corresponding statement true for perverse sheaves smooth along a stratification given by affine spaces? The latter is certainly a highest weight category, computations in it can be undertaken topologically, etc. So as a starting point I would like to understand the topological reason for its truth for the "toy example".

  2. In the same vein as 1) I would like to know whether this truly is a "geometric" fact, i.e., does it hold if I consider my sheaves with coefficients in a commutative ring say?

  3. Computing extension groups of Verma modules is an old problem. If there is any hope for doing this topologically, I would think a reasonable place to start would be to compute $\mathrm{Ext}^0$ topologically!

  4. In the same vein as 3). One can see that the extensions of Verma modules is given by (compactly supported) cohomology (appropriately shifted) of intersections of Schubert cells with opposite Schubert cells. This is related to my earlier questions:

Intersection of plus/minus cells in Bialynicki-Birula decomposition

A cohomology computation request.

  • The above fact about homomorphisms between Vermas translates to the lowest non-vanishing cohomology (compactly supported) being one dimensional. These are smooth affine varieties, but (at least in low ranks) their Betti numbers satisfy a curious "Poincare duality"/palindromic type phenomenon. This phenomenon is even more starkly visible if one further looks at the Hodge numbers. Amusingly, since these varieties are smooth and irreducible, one immediately gets that the highest non-vanishing extension group (when it is possible to have morphisms between the Vermas) is one dimensional and concentrated in the "right" degree. This latter fact can also be shown algebraically, but requires a careful argument using translation functors (which can also be done geometrically without ever knowing anything about the intersections, but now I am digressing). Anyway, a topological reason as in my question may hopefully give some insight as to whether this palindromic phenomenon is a low rank coincidence or has any hope for holding in general.

Apologies if any of the reasons above are too vague/ranting, I didn't want to throw in all of that in my original question in case the answer was something blatantly obvious that I had been overlooking.

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    $\begingroup$ The dimension bound is elementary and occurs in Verma's thesis, with no reference to later ideas about the BGG category or its blocks. While a formulation and proof in the geometric setting is probably possible, is there any reason to expect such a proof to provide extra insight? In other words, could you add some motivation for the question? It looks like using a sophisticated theory to attack an easy algebraic problem. (In contrast, the composition factor multiplicities in Verma modules really need such a theory.) $\endgroup$ – Jim Humphreys Dec 14 '12 at 14:48
  • $\begingroup$ Good question! I remember talking to Soergel about this a while ago. If memory serves me right he told me that this is no longer true on partial flag varieties, but unfortunately I can't remember where the first counterexample occurs. Also I have often wondered about torsion in the cohomology of these intersections, but I haven't gotten beyond wondering... (some motivation for why one might care is in the section on R-varieties in arxiv.org/abs/1209.3760). $\endgroup$ – Geordie Williamson Dec 16 '12 at 2:50
  • $\begingroup$ Deleted a previous comment of mine where I thought I had an argument, since it was a pipe dream. $\endgroup$ – Reladenine Vakalwe Dec 17 '12 at 1:56
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    $\begingroup$ As the Mathoverflow bot has so graciously pinged this question, I may as well add the following. The "curious Poincare duality" and "palindromic phenomenon" mentioned above has a high powered explanation. Namely: Koszul duality. This can in turn be used to show the dimension bound asked for. But this is using a blowtorch to light a candle. $\endgroup$ – Reladenine Vakalwe Dec 31 '12 at 2:25

It may be helpful to look at the 1987 paper by Kari Vilonen and his student Ren Mirollo here. They study BGG reciprocity on perverse sheaves in some generality, working over a field. Here the intermediate (or standard) objects corresponding to Verma modules in the classical BGG category are straightforward to define, using an assumed Whitney stratification like that of the flag variety. The functors $\mathrm{Ext}^i$ then come into play.

Taken in isolation, the original Verma modules are easy enough to construct: they played a technical role for Chevalley and Harish-Chandra in getting uniform existence proofs for finite dimensional simple modules in the Lie algebra framework. But arbitrary simple quotients of Verma modules are quite elusive, requiring deeper ideas. This shows up again in the setting of perverse sheaves.

As in your question, Mirollo-Vilonen were proceeding in a somewhat speculative way. To work over suitable rings, you'd probably have to go further into the work of Mirkovic-Vilonen on geometric Langlands.

ADDED: Concerning Geordie's comment on the case of parabolic Verma modules (relative to "partial flag varieties"), it's clear from the algebraic study already done that Hom spaces get much more complicated than for ordinary Verma modules. (See Section 9.10 of my 2008 AMS book on the BGG category for some details and references.) As far as I know, none of this has been effectively translated into the language of perverse sheaves along the lines of Mirollo-Vilonen.


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