This is a companion paper of [BFGT, BFT]. We prove an equivalence relating representations of a d... more This is a companion paper of [BFGT, BFT]. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of twisted Sp(2n, C[[t]])-equivariant D-modules on the so called mirabolic affine Grassmannian of Sp(2n). We also discuss (conjectural) extension of this equivalence to the case of quantum supergroups and to some exceptional supergroups.
The set of orbits of GL(V) in F l(V)×F l(V)×V is finite, and is parametrized by the set of certai... more The set of orbits of GL(V) in F l(V)×F l(V)×V is finite, and is parametrized by the set of certain decorated permutations in a work of Magyar, Weyman, Zelevinsky. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of GL(V) arising from F l(V) × F l(V) × V. We also give conjectural applications to the classification of unipotent mirabolic character sheaves on GL(V) × V .
We propose a construction of the Coulomb branch of a $3d\ {\mathcal N}=4$ gauge theory correspond... more We propose a construction of the Coulomb branch of a $3d\ {\mathcal N}=4$ gauge theory corresponding to a choice of a connected reductive group $G$ and a symplectic finite-dimensional reprsentation $\mathbf M$ of $G$, satisfying certain anomaly cancellation condition. This extends the construction of arXiv:1601.03586 (where it was assumed that ${\mathbf M}={\mathbf N}\oplus{\mathbf N}^*$ for some representation $\mathbf N$ of $G$). Our construction goes through certain "universal" ring object in the twisted derived Satake category of the symplectic group $Sp(2n)$. The construction of this object uses a categorical version of the Weil representation; we also compute the image of this object under the (twisted) derived Satake equivalence and show that it can be obtained from the theta-sheaf introduced by S.Lysenko on $\operatorname{Bun}_{Sp(2n)}({\mathbb P}^1)$ via certain Radon transform. We also discuss applications of our construction to a potential mathematical construct...
This is a successive paper of [BFGT]. We prove an equivalence between the category of finite-dime... more This is a successive paper of [BFGT]. We prove an equivalence between the category of finite-dimensional representations of degenerate supergroup $\underline{GL}(M|N)$ and the category of $(GL_M(O) \ltimes U_{M, N}(F), χ_{M, N})$-equivariant D-modules on $Gr_{N}$. We also prove that we can realize the category of finite-dimensional representations of degenerate supergroup $\underline{GL}(M|N)$ as a category of D-modules on the mirabolic subgroup $Mir_L(F)$ with certain equivariant conditions for any $L$ bigger than $N$ and $M$.
The set of orbits of GL(V ) in F l(V ) × F l(V ) × V is finite, and is parametrized by the set of... more The set of orbits of GL(V ) in F l(V ) × F l(V ) × V is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of GL(V ) arising from F l(V )×F l(V )×V . We also give conjectural applications to the classification of unipotent mirabolic character sheaves on GL(V )× V .
We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalen... more We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup with the category of GL(N-1,ℂ[[t]])-equivariant perverse sheaves on the affine Grassmannian of GL_N. We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.
Let G be a semisimple, simply connected algebraic group over an algebraically closed field of cha... more Let G be a semisimple, simply connected algebraic group over an algebraically closed field of characteristic zero. We prove that the ∞-category of D-modules on the loop group of G is equivalent to the monoidal colimit of the ∞-categories of D-modules on the standard parahoric subgroups. This also follows from arXiv:2009.10998, but the present paper gives a simpler proof. The idea is to develop a combinatorial model for the path space of a simplicial complex, in which 'paths' are sequences of adjacent simplices, and to use a generalized version of hyperdescent for D-modules. We also give two more applications of this hyperdescent theorem: triviality of D-modules on the 'schematic Bruhat-Tits building,' which was first established by Varshavsky using a different method, and triviality of D-modules on the `simplicial affine Springer resolution.'
For a smooth variety X over an algebraically closed field of characteristic p, to a differential ... more For a smooth variety X over an algebraically closed field of characteristic p, to a differential 1-form α on the Frobenius twist X^(1) one can associate an Azumaya algebra D_X,α, defined as a certain central reduction of the algebra D_X of "crystalline differential operators" on X. For a resolution of singularities π:X→ Y of an affine variety Y, we study for which α does the class [ D_X,α] in the Brauer group Br(X^(1)) descend to Y^(1). In the case when X is symplectic, this question is related to Fedosov quantizations in characteristic p and the construction of non-commutative resolutions of Y. We prove that the classes [ D_X,α] descend étale locally for all α if O_Y≃π_* O_X and R^1,2π_* O_X =0. We also define a certain class of resolutions which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic 0 to an algebraically closed field of characteristic p classes [ D_X,α] descend to Y^(1) global...
The set of orbits of GL(V) in Fl(V)× Fl(V)× V is finite, and is parametrized by the set of certai... more The set of orbits of GL(V) in Fl(V)× Fl(V)× V is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of GL(V) arising from Fl(V)× Fl(V)× V. We also give conjectural applications to the classification of unipotent mirabolic character sheaves on GL(V)× V.
This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations o... more This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of SO(N-1,C[[t]])-equivariant perverse sheaves on the affine Grassmannian of SO_N. We explain how this equivalence fits into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.
For a smooth variety X over an algebraically closed field of characteristic p, to a differential ... more For a smooth variety X over an algebraically closed field of characteristic p, to a differential 1-form α on the Frobenius twist X^(1) one can associate an Azumaya algebra D_X,α, defined as a certain central reduction of the algebra D_X of "crystalline differential operators" on X. For a resolution of singularities π:X→ Y of an affine variety Y, we study for which α does the class [ D_X,α] in the Brauer group Br(X^(1)) descend to Y^(1). In the case when X is symplectic, this question is related to Fedosov quantizations in characteristic p and the construction of non-commutative resolutions of Y. We prove that the classes [ D_X,α] descend étale locally for all α if O_Y≃π_* O_X and R^1,2π_* O_X =0. We also define a certain class of resolutions which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic 0 to an algebraically closed field of characteristic p classes [ D_X,α] descend to Y^(1) global...
In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable sys... more In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable system by showing that the global sections of critically twisted differential operators on the moduli stack of G-bundles on an algebraic curve is identified with the ring of regular functions on the space of G-opers; they deduce existence of an automorphic D-module corresponding to a local system carrying a structure of an oper. In this note we show for G=GL(n) that those results admit a short proof by reduction to positive characteristic, where the result is deduced from generic Langlands duality established earlier by the first author and A. Braverman. The appendix contains a proof of some properties of the p-curvature map restricted to the space of opers.
This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations o... more This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of $SO(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $SO_N$. We explain how this equivalence fits into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.
In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable sys... more In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable system by showing that the global sections of critically twisted differential operators on the moduli stack of G-bundles on an algebraic curve is identified with the ring of regular functions on the space of G-opers; they deduce existence of an automorphic D-module corresponding to a local system carrying a structure of an oper. In this note we show for G=GL(n) that those results admit a short proof by reduction to positive characteristic, where the result is deduced from generic Langlands duality established earlier by the first author and A. Braverman. The appendix contains a proof of some properties of the p-curvature map restricted to the space of opers.
Let $G$ be a semisimple, simply connected algebraic group over an algebraically closed field of c... more Let $G$ be a semisimple, simply connected algebraic group over an algebraically closed field of characteristic zero. We prove that the $\infty$-category of D-modules on the loop group of $G$ is equivalent to the monoidal colimit of the $\infty$-categories of D-modules on the standard parahoric subgroups. This also follows from arXiv:2009.10998, but the present paper gives a simpler proof. The idea is to develop a combinatorial model for the path space of a simplicial complex, in which 'paths' are sequences of adjacent simplices, and to use a generalized version of hyperdescent for D-modules. We also give two more applications of this hyperdescent theorem: triviality of D-modules on the 'schematic Bruhat-Tits building,' which was first established by Varshavsky using a different method, and triviality of D-modules on the `simplicial affine Springer resolution.'
We prove D. Gaiotto’s conjecture about geometric Satake equivalence for quantum supergroup Uq(gl(... more We prove D. Gaiotto’s conjecture about geometric Satake equivalence for quantum supergroup Uq(gl(N − 1|N)) for generic q. The equivalence goes through the category of factorizable sheaves.
Given a linear category over a finite field such that the moduli space of its objects is a smooth... more Given a linear category over a finite field such that the moduli space of its objects is a smooth Artin stack (and some additional conditions) we give formulas for an exponential sum over the set of absolutely indecomposable objects and a stacky sum over the set of all objects of the category, respectively, in terms of the geometry of the cotangent bundle on the moduli stack. The first formula was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas. It provides a new approach for counting absolutely indecomposable quiver representations, vector bundles with parabolic structure on a projective curve, and irreducible etale local systems (via a result of Deligne). Our second formula resembles formulas appearing in the theory of Donaldson-Thomas invariants.
In [De2], Deligne showed that the reduced lift presentation of a finite type generalized braid gr... more In [De2], Deligne showed that the reduced lift presentation of a finite type generalized braid group remains correct if it is (suitably) interpreted as a presentation of a topological monoid. In this expository paper, we point out that Deligne’s argument does not require the ‘finite type’ hypothesis, so it gives a different proof of [Do, Thm. 5.1]. We also review how to use this result to construct an action of the braid group on the finite or affine Hecke ∞-category via intertwining functors.
We compute the Frobenius trace functions of mirabolic character sheaves defined over a finite fie... more We compute the Frobenius trace functions of mirabolic character sheaves defined over a finite field. The answer is given in terms of the character values of general linear groups over the finite field, and the structure constants of multiplication in the mirabolic Hall-Littlewood basis of symmetric functions, introduced by Shoji.
Let $G$ be a semisimple simply-connected algebraic group over an algebraically closed field of ch... more Let $G$ be a semisimple simply-connected algebraic group over an algebraically closed field of characteristic zero. We prove that the affine Hecke category associated to $G$ is equivalent to the colimit, evaluated in the $\infty$-category of stable monoidal $\infty$-categories, of the finite Hecke subcategories associated to standard parahoric subgroups of $G$. The same method yields analogous colimit presentations of the 0-Hecke monoid (where the colimit is taken in monoids in spaces) and of the Hecke algebra (where the colimit is taken in DG-algebras). We also construct a strong monoidal functor from any finite or affine type braid group to the corresponding Hecke category.
This is a companion paper of [BFGT, BFT]. We prove an equivalence relating representations of a d... more This is a companion paper of [BFGT, BFT]. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of twisted Sp(2n, C[[t]])-equivariant D-modules on the so called mirabolic affine Grassmannian of Sp(2n). We also discuss (conjectural) extension of this equivalence to the case of quantum supergroups and to some exceptional supergroups.
The set of orbits of GL(V) in F l(V)×F l(V)×V is finite, and is parametrized by the set of certai... more The set of orbits of GL(V) in F l(V)×F l(V)×V is finite, and is parametrized by the set of certain decorated permutations in a work of Magyar, Weyman, Zelevinsky. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of GL(V) arising from F l(V) × F l(V) × V. We also give conjectural applications to the classification of unipotent mirabolic character sheaves on GL(V) × V .
We propose a construction of the Coulomb branch of a $3d\ {\mathcal N}=4$ gauge theory correspond... more We propose a construction of the Coulomb branch of a $3d\ {\mathcal N}=4$ gauge theory corresponding to a choice of a connected reductive group $G$ and a symplectic finite-dimensional reprsentation $\mathbf M$ of $G$, satisfying certain anomaly cancellation condition. This extends the construction of arXiv:1601.03586 (where it was assumed that ${\mathbf M}={\mathbf N}\oplus{\mathbf N}^*$ for some representation $\mathbf N$ of $G$). Our construction goes through certain "universal" ring object in the twisted derived Satake category of the symplectic group $Sp(2n)$. The construction of this object uses a categorical version of the Weil representation; we also compute the image of this object under the (twisted) derived Satake equivalence and show that it can be obtained from the theta-sheaf introduced by S.Lysenko on $\operatorname{Bun}_{Sp(2n)}({\mathbb P}^1)$ via certain Radon transform. We also discuss applications of our construction to a potential mathematical construct...
This is a successive paper of [BFGT]. We prove an equivalence between the category of finite-dime... more This is a successive paper of [BFGT]. We prove an equivalence between the category of finite-dimensional representations of degenerate supergroup $\underline{GL}(M|N)$ and the category of $(GL_M(O) \ltimes U_{M, N}(F), χ_{M, N})$-equivariant D-modules on $Gr_{N}$. We also prove that we can realize the category of finite-dimensional representations of degenerate supergroup $\underline{GL}(M|N)$ as a category of D-modules on the mirabolic subgroup $Mir_L(F)$ with certain equivariant conditions for any $L$ bigger than $N$ and $M$.
The set of orbits of GL(V ) in F l(V ) × F l(V ) × V is finite, and is parametrized by the set of... more The set of orbits of GL(V ) in F l(V ) × F l(V ) × V is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of GL(V ) arising from F l(V )×F l(V )×V . We also give conjectural applications to the classification of unipotent mirabolic character sheaves on GL(V )× V .
We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalen... more We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup with the category of GL(N-1,ℂ[[t]])-equivariant perverse sheaves on the affine Grassmannian of GL_N. We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.
Let G be a semisimple, simply connected algebraic group over an algebraically closed field of cha... more Let G be a semisimple, simply connected algebraic group over an algebraically closed field of characteristic zero. We prove that the ∞-category of D-modules on the loop group of G is equivalent to the monoidal colimit of the ∞-categories of D-modules on the standard parahoric subgroups. This also follows from arXiv:2009.10998, but the present paper gives a simpler proof. The idea is to develop a combinatorial model for the path space of a simplicial complex, in which 'paths' are sequences of adjacent simplices, and to use a generalized version of hyperdescent for D-modules. We also give two more applications of this hyperdescent theorem: triviality of D-modules on the 'schematic Bruhat-Tits building,' which was first established by Varshavsky using a different method, and triviality of D-modules on the `simplicial affine Springer resolution.'
For a smooth variety X over an algebraically closed field of characteristic p, to a differential ... more For a smooth variety X over an algebraically closed field of characteristic p, to a differential 1-form α on the Frobenius twist X^(1) one can associate an Azumaya algebra D_X,α, defined as a certain central reduction of the algebra D_X of "crystalline differential operators" on X. For a resolution of singularities π:X→ Y of an affine variety Y, we study for which α does the class [ D_X,α] in the Brauer group Br(X^(1)) descend to Y^(1). In the case when X is symplectic, this question is related to Fedosov quantizations in characteristic p and the construction of non-commutative resolutions of Y. We prove that the classes [ D_X,α] descend étale locally for all α if O_Y≃π_* O_X and R^1,2π_* O_X =0. We also define a certain class of resolutions which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic 0 to an algebraically closed field of characteristic p classes [ D_X,α] descend to Y^(1) global...
The set of orbits of GL(V) in Fl(V)× Fl(V)× V is finite, and is parametrized by the set of certai... more The set of orbits of GL(V) in Fl(V)× Fl(V)× V is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of GL(V) arising from Fl(V)× Fl(V)× V. We also give conjectural applications to the classification of unipotent mirabolic character sheaves on GL(V)× V.
This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations o... more This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of SO(N-1,C[[t]])-equivariant perverse sheaves on the affine Grassmannian of SO_N. We explain how this equivalence fits into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.
For a smooth variety X over an algebraically closed field of characteristic p, to a differential ... more For a smooth variety X over an algebraically closed field of characteristic p, to a differential 1-form α on the Frobenius twist X^(1) one can associate an Azumaya algebra D_X,α, defined as a certain central reduction of the algebra D_X of "crystalline differential operators" on X. For a resolution of singularities π:X→ Y of an affine variety Y, we study for which α does the class [ D_X,α] in the Brauer group Br(X^(1)) descend to Y^(1). In the case when X is symplectic, this question is related to Fedosov quantizations in characteristic p and the construction of non-commutative resolutions of Y. We prove that the classes [ D_X,α] descend étale locally for all α if O_Y≃π_* O_X and R^1,2π_* O_X =0. We also define a certain class of resolutions which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic 0 to an algebraically closed field of characteristic p classes [ D_X,α] descend to Y^(1) global...
In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable sys... more In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable system by showing that the global sections of critically twisted differential operators on the moduli stack of G-bundles on an algebraic curve is identified with the ring of regular functions on the space of G-opers; they deduce existence of an automorphic D-module corresponding to a local system carrying a structure of an oper. In this note we show for G=GL(n) that those results admit a short proof by reduction to positive characteristic, where the result is deduced from generic Langlands duality established earlier by the first author and A. Braverman. The appendix contains a proof of some properties of the p-curvature map restricted to the space of opers.
This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations o... more This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of $SO(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $SO_N$. We explain how this equivalence fits into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.
In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable sys... more In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable system by showing that the global sections of critically twisted differential operators on the moduli stack of G-bundles on an algebraic curve is identified with the ring of regular functions on the space of G-opers; they deduce existence of an automorphic D-module corresponding to a local system carrying a structure of an oper. In this note we show for G=GL(n) that those results admit a short proof by reduction to positive characteristic, where the result is deduced from generic Langlands duality established earlier by the first author and A. Braverman. The appendix contains a proof of some properties of the p-curvature map restricted to the space of opers.
Let $G$ be a semisimple, simply connected algebraic group over an algebraically closed field of c... more Let $G$ be a semisimple, simply connected algebraic group over an algebraically closed field of characteristic zero. We prove that the $\infty$-category of D-modules on the loop group of $G$ is equivalent to the monoidal colimit of the $\infty$-categories of D-modules on the standard parahoric subgroups. This also follows from arXiv:2009.10998, but the present paper gives a simpler proof. The idea is to develop a combinatorial model for the path space of a simplicial complex, in which 'paths' are sequences of adjacent simplices, and to use a generalized version of hyperdescent for D-modules. We also give two more applications of this hyperdescent theorem: triviality of D-modules on the 'schematic Bruhat-Tits building,' which was first established by Varshavsky using a different method, and triviality of D-modules on the `simplicial affine Springer resolution.'
We prove D. Gaiotto’s conjecture about geometric Satake equivalence for quantum supergroup Uq(gl(... more We prove D. Gaiotto’s conjecture about geometric Satake equivalence for quantum supergroup Uq(gl(N − 1|N)) for generic q. The equivalence goes through the category of factorizable sheaves.
Given a linear category over a finite field such that the moduli space of its objects is a smooth... more Given a linear category over a finite field such that the moduli space of its objects is a smooth Artin stack (and some additional conditions) we give formulas for an exponential sum over the set of absolutely indecomposable objects and a stacky sum over the set of all objects of the category, respectively, in terms of the geometry of the cotangent bundle on the moduli stack. The first formula was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas. It provides a new approach for counting absolutely indecomposable quiver representations, vector bundles with parabolic structure on a projective curve, and irreducible etale local systems (via a result of Deligne). Our second formula resembles formulas appearing in the theory of Donaldson-Thomas invariants.
In [De2], Deligne showed that the reduced lift presentation of a finite type generalized braid gr... more In [De2], Deligne showed that the reduced lift presentation of a finite type generalized braid group remains correct if it is (suitably) interpreted as a presentation of a topological monoid. In this expository paper, we point out that Deligne’s argument does not require the ‘finite type’ hypothesis, so it gives a different proof of [Do, Thm. 5.1]. We also review how to use this result to construct an action of the braid group on the finite or affine Hecke ∞-category via intertwining functors.
We compute the Frobenius trace functions of mirabolic character sheaves defined over a finite fie... more We compute the Frobenius trace functions of mirabolic character sheaves defined over a finite field. The answer is given in terms of the character values of general linear groups over the finite field, and the structure constants of multiplication in the mirabolic Hall-Littlewood basis of symmetric functions, introduced by Shoji.
Let $G$ be a semisimple simply-connected algebraic group over an algebraically closed field of ch... more Let $G$ be a semisimple simply-connected algebraic group over an algebraically closed field of characteristic zero. We prove that the affine Hecke category associated to $G$ is equivalent to the colimit, evaluated in the $\infty$-category of stable monoidal $\infty$-categories, of the finite Hecke subcategories associated to standard parahoric subgroups of $G$. The same method yields analogous colimit presentations of the 0-Hecke monoid (where the colimit is taken in monoids in spaces) and of the Hecke algebra (where the colimit is taken in DG-algebras). We also construct a strong monoidal functor from any finite or affine type braid group to the corresponding Hecke category.
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Papers by Roman Travkin