Parabolic convolution of perverse sheaves in terms of the Hecke algebra

Question

It is "well-known" that the Hecke algebra $\mathcal{H}$ can be thought of as the Grothendieck group for the category of perverse sheaves on $G/B$, where the product in $\mathcal{H}$ corresponds to convolution of sheaves by the Borel subgroup. This means, given perverse sheaves $X$ and $Y$ on $G/B$ and their classes $[X]$ and $[Y]$ in the Grothendieck group considered as elements of $\mathcal{H}$, the element $[X][Y]$ is (the class of) the sheaf

$$q_B(\pi^*(X)\boxtimes Y),$$

where $\pi$ is the quotient map $G\rightarrow G/B$ and $q_B$ is the operation that takes a sheaf on $G\times G/B$ which is equivariant under the action of $B$ by $b\cdot (g, hB) = (gb^{-1}, bhB)$ and quotients out by this action, mapping to a sheaf on $G/B$.

Now I want to think about convolution using a parabolic subgroup larger than the Borel. Suppose $P$ is a parabolic, $X$ is a perverse sheaf on $G/B$ which is pulled back from $G/P$, and $Y$ is $P$-equivariant.

What is the class of

$$q_P(\pi^*(X)\boxtimes Y),$$

where $q_P$ now quotients by the action of $P$ by $p\cdot (g, hB) = (gp^{-1}, pHB)$ in terms of the Hecke algebra?

Motivation: Actually, I only care about the case where $X$ and $Y$ are IC sheaves of Schubert varieties, so $[X]$ and $[Y]$ are Kazhdan--Lusztig basis elements. In this case, by the Decomposition Theorem the product will give a positive, bar--invariant sum of Kazhdan--Lusztig basis elements, so if you find good choices of $X$ and $Y$ and understand this product, you get a good inductive method for calculating Kazhdan--Lusztig polynomials. In some cases I am interested in, I have calculated the product geometrically by localization, but sticking to algebra would make things cleaner and likely easier to write up in general. There are cases in which this has been done, most notably Polo's paper showing that any polynomial with positive integer coefficients and constant term 1 is a Kazhdan--Lusztig polynomial; an answer to this question should allow one to dispense with geometry in his paper and formulate his calculations entirely within the Hecke algebra (given a theorem that says the algebraic analogue of parabolic convolution produces positive sums of Kazhdan--Lusztig elements).

Although everything makes sense for an arbitrary Kac-Moody group, I am happy with an answer for the finite dimensional case, or even with answers for type A.

Pre-emptive requests: (1) I expect parabolic Kazhdan--Lusztig elements will come up. As there are several variants, please tell us which one you mean, with reference to a paper using your notation. (2) As you may have noted from my vague description of $q_B$ and any other mistakes I might have made above, I don't really understand perverse sheaves and intersection cohomology. Please feel free to correct me, and please give me an answer I can understand.

Answer 1

Let $G$ be a connected reductive algebraic group (over $\mathbb{C}$) and fix a Borel subgroup $B \subset G$. One can consider the 2-category with objects parabolic subgroups $P \supset B$ and 1-morphisms $P \to Q$ given by $D^b_{P\times Q}(G)$ (the $P \times Q$-equivariant derived category of $G$ with respect to the action $(p,q) \cdot g = pgq^{-1}$ for $p \in P$, $q \in Q$ and $g \in G$). 2-morphisms are the morphisms of $D^b_{P \times Q}(G)$. Composition of 1-morphisms is given by convolution:

$* : D_{P \times Q}(G) \times D_{Q \times R}(G) \to D_{P \times R}(G)$

(you can probably guess how this is defined using the description you give above).

Then your question is a special case of the following question:

Q: describe the Grothendieck group of this 2-category (a 1-category).

The answer is that Grothendieck group is* what I call the Schur algebroid (for want of a better name). See the beginning of my paper "Singular Soergel bimodules" on the arXiv. (I am only referring to my paper because it is a convenient reference. Certainly these things have been known to experts since at least the early 80's.)

If you would prefer not to look at this paper here is a direct answer to your question. For any subset $I \subset S$ of the simple reflections let $\underline{H}_I$ denote the Kazhdan-Lusztig basis element indexed by the longest element of the standard parabolic $W_I$. Then, given $h \in \mathcal{H}\underline{H}_I$ and $h' \in \underline{H}_I \mathcal{H}$ one can define

$h*_I h' := \frac{1}{\pi(I)} hh'$

where $\pi(I)$ denotes the Poincaré polynomial of $W_I$. (Note that this element really lives in the $\mathbb{Z}[v,v^{-1}]$ form of the Hecke algebra.) This is the class in the Hecke algebra you are looking for.

*: Of course this is not true because one has lost the $q$: what I really mean is that one should either consider the split Grothendieck group of semi-simple complexes (or parity sheaves if one prefers) or use an appropriate mixed version.

Final note: you talk about Polo's result about arbitrary polynomials $ \in 1 + q\mathbb{N}[q]$ occuring as KL-polynomials. I recall that there is a purely combinatorial proof of this result in the literature. Unfortunately I can't remember the title or author, but it shouldn't be difficult to find.