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Certain Kazhdan-Lusztig polynomials compute the so-called Brylinski-Kostant filtration on weight spaces of irreducible representations. They also compute multiplicities of irreducible modules occurring in global sections of line bundles on cotangent bundles of flag varieties. These two ideas are related.

In more detail: Let $G$ be a semisimple algebraic group over $\mathbb C$. Fix a Borel subgroup $B \subseteq G$ with unipotent radical $U$ and choose a principal nilpotent element $X \in \textrm{Lie}(U)$ (a principal nilpotent element is one whose $G$-orbit is dense in the subvariety of all nilpotent elements). Let $\lambda, \mu$ be dominant weights and consider the $\mu$-weight space $V_\mu(\lambda)$ of the irreducible representation $V(\lambda)$ of $G$. Then there is a filtration -- the Brylinski-Kostant filtration -- on $V_\mu(\lambda)$ coming from the action of $X$ on $V(\lambda)$: define $\mathcal F^n( V_\mu(\lambda) )$ to be the subspace of $V_\mu(\lambda)$ consisting of vectors killed by n+1 applications of $X$. Since $X$ is nilpotent, this does indeed give a filtration on $V_\mu(\lambda)$.

Now, an important theorem due to R. Brylinski (1) is that certain Kazhdan-Lusztig polynomials give the dimensions of the subspaces in this filtration. This is a very interesting theorem: to prove it, Brylinski actually proves an intermediate geometric theorem relating Kazhdan-Lusztig polynomials to degrees of twisted functions on twisted versions of the "base affine space" G/U G/T (which are connected to degrees of sections of line bundles on the cotangent bundle of G/B), so that along the way she gives yet another interpretation of Kazhdan-Lusztig polynomials. In particular, they compute multiplicities of irreducible modules occurring in global sections of line bundles on the cotangent bundle of G/B. They also give information on the degrees of sections of these bundles (I will just refer to Brylinski's paper for the appropriate definition of "degree").

Here I will humbly submit my own work: in (2) I extended some of Brylinski's results to the case where the nilpotent element is not necessarily principal. In this case, certain Kazhdan-Lusztig polynomials also appear in an analogous filtration. Further, the full cotangent bundle of G/B is replaced by an appropriate subbundle E of the cotangent bundle, and one has the following result: certain Kazhdan-Lusztig polynomials compute multiplicities of irreducibles occurring in global sections of line bundles on E. (Unfortunately the full general statement that one would like to make is still a conjecture, due to difficult technical issues involving cohomology vanishing of these bundles).

Remark: Both Brylinskis have made fundamental contributions to the theory of Kazhdan-Lusztig polynomials. Jean-Luc Brylinski and M. Kashiwara, and independently Beilinson and Bernstein, proved the Kazhdan-Lusztig conjectures (this is the interpretation mentioned by Jan in his question); Ranee Brylinski, Jean-Luc's wife, gave the interpretation I've described above.

References:

(1) Brylinski, R. K. Limits of weight spaces, Lusztig's $q$-analogs, and fiberings of adjoint orbits. J. Amer. Math. Soc., 1989, Vol. 3, pp. 517--533.

(2) Available on the ArXiV here.

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Certain Kazhdan-Lusztig polynomials compute the so-called Brylinski-Kostant filtration on weight spaces of irreducible representations. They also compute multiplicities of irreducible modules occurring in global sections of line bundles on cotangent bundles of flag varieties. These two ideas are related.

In more detail: Let $G$ be a semisimple algebraic group over $\mathbb C$. Fix a Borel subgroup $B \subseteq G$ with unipotent radical $U$ and choose a principal nilpotent element $X \in \textrm{Lie}(U)$ (a principal nilpotent element is one whose $G$-orbit is dense in the subvariety of all nilpotent elements). Let $\lambda, \mu$ be dominant weights and consider the $\mu$-weight space $V_\mu(\lambda)$ of the irreducible representation $V(\lambda)$ of $G$. Then there is a filtration -- the Brylinski-Kostant filtration -- on $V_\mu(\lambda)$ coming from the action of $X$ on $V(\lambda)$: define $\mathcal F^n( V_\mu(\lambda) )$ to be the subspace of $V_\mu(\lambda)$ consisting of vectors killed by n+1 applications of $X$. Since $X$ is nilpotent, this does indeed give a filtration on $V_\mu(\lambda)$.

Now, an important theorem due to R. Brylinski (1) is that certain Kazhdan-Lusztig polynomials give the dimensions of the subspaces in this filtration. This is a very interesting theorem: to prove it, Brylinski actually proves an intermediate geometric theorem relating Kazhdan-Lusztig polynomials to degrees of functions on twisted versions of the "base affine space" G/U (which are connected to degrees of sections of line bundles on the cotangent bundle of G/B), so that along the way she gives yet another interpretation of Kazhdan-Lusztig polynomials. In particular, they compute multiplicities of irreducible modules occurring in global sections of line bundles on the cotangent bundle of G/B. They also give information on the degrees of sections of these bundles (I will just refer to Brylinski's paper for the appropriate definition of "degree").

Here I will humbly submit my own work: in (2) I extended some of Brylinski's results to the case where the nilpotent element is not necessarily principal. In this case, certain Kazhdan-Lusztig polynomials also appear in an analogous filtration. Further, the full cotangent bundle of G/B is replaced by an appropriate subbundle E of the cotangent bundle, and one has the following result: certain Kazhdan-Lusztig polynomials compute multiplicities of irreducibles occurring in global sections of line bundles on E. (Unfortunately the full general statement that one would like to make is still a conjecture, due to difficult technical issues involving cohomology vanishing of these bundles).

Remark: Both Brylinskis have made fundamental contributions to the theory of Kazhdan-Lusztig polynomials. Jean-Luc Brylinski and M. Kashiwara, and independently Beilinson and Bernstein, proved the Kazhdan-Lusztig conjectures (this is the interpretation mentioned by Jan in his question); Ranee Brylinski, Jean-Luc's wife, gave the interpretation I've described above.

References:

(1) Brylinski, R. K. Limits of weight spaces, Lusztig's $q$-analogs, and fiberings of adjoint orbits. J. Amer. Math. Soc., 1989, Vol. 3, pp. 517--533.

(2) Available on the ArXiV here.

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Certain Kazhdan-Lusztig polynomials compute the so-called Brylinski-Kostant filtration on weight spaces of irreducible representations. They also compute multiplicities of irreducible modules occurring in global sections of line bundles on cotangent bundles of flag varieties. These two ideas are related.

In more detail: Let $G$ be a semisimple algebraic group over $\mathbb C$. Fix a Borel subgroup $B \subseteq G$ with unipotent radical $U$ and choose a principal nilpotent element $X \in \textrm{Lie}(U)$ (a principal nilpotent element is one whose $G$-orbit is dense in the subvariety of all nilpotent elements). Let $\lambda, \mu$ be dominant weights and consider the $\mu$-weight space $V_\mu(\lambda)$ of the irreducible representation $V(\lambda)$ of $G$. Then there is a filtration -- the Brylinski-Kostant filtration -- on $V_\mu(\lambda)$ coming from the action of $X$ on $V(\lambda)$: define $\mathcal F^n( V_\mu(\lambda) )$ to be the subspace of $V_\mu(\lambda)$ consisting of vectors killed by n+1 applications of $X$. Since $X$ is nilpotent, this does indeed give a filtration on $V_\mu(\lambda)$.

Now, an important theorem due to R. Brylinski (1) is that certain Kazhdan-Lusztig polynomials give the dimensions of the subspaces in this filtration. This is a very interesting theorem: to prove it, Brylinski actually proves an intermediate geometric theorem relating Kazhdan-Lusztig polynomials to degrees of functions on twisted versions of the "base affine space" G/U (which are connected to degrees of sections of line bundles on the cotangent bundle of G/B), so that along the way she gives yet another interpretation of Kazhdan-Lusztig polynomials. In particular, they compute multiplicities of irreducible modules occurring in global sections of line bundles on the cotangent bundle of G/B. They also give information on the degrees of sections of these bundles (I will just refer to Brylinski's paper for the appropriate definition of "degree").

Here I will humbly submit my own work: in (2) I extended some of Brylinski's results to the case where the nilpotent element is not necessarily principal. In this case, certain Kazhdan-Lusztig polynomials also appear in an analogous filtration. Further, the full cotangent bundle of G/B is replaced by an appropriate subbundle E of the cotangent bundle, and one has the following result: certain Kazhdan-Lusztig polynomials compute multiplicities of irreducibles occurring in global sections of line bundles on E. (Unfortunately the full general statement that one would like to make is still a conjecture, due to difficult technical issues involving cohomology vanishing of these bundles).

References:

(1) Brylinski, R. K. Limits of weight spaces, Lusztig's $q$-analogs, and fiberings of adjoint orbits. J. Amer. Math. Soc., 1989, Vol. 3, pp. 517--533.

(2) Available on the ArXiV here.

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