TY  - GEN
AB  - Losev introduced the scheme X of almost commuting elements (i.e., elements commuting upto a rank one element) of g = sp(V) for a symplectic vector space V and discussed its algebro-geometric properties. We construct a Lagrangian subscheme Xnil of X and show that it is a complete intersection of dimension dim(g) + 1/2 dim(V ) and compute its irreducible components.

We study the quantum Hamiltonian reduction of the algebra D(g) of differential operators on the Lie algebra g tensored with the Weyl algebra with respect to the action of the symplectic group, and show that it is isomorphic to the spherical subalgebra of a certain rational Cherednik algebra of Type C. We contruct a category Cc of D-modules whose characteristic variety is contained in Xnil and construct an exact functor from this category to the category O of the above rational Cherednik algebra. Simple objects of the category Cc are mirabolic analogs of Lusztig's character sheaves.

We also define and study a group-theoretic version of Losev’s almost commuting scheme as well as the above quantum Hamiltonian reduction problem.
AD  - University of Chicago
AU  - Goyal, Pallav
DA  - 2023-06
DO  - 10.6082/uchicago.6424
DO  - doi
ED  - Victor Ginzburg
ED  - Victor Ginzburg
ED  - Alexander Beilinson
ID  - 6424
KW  - Mathematics
KW  - Cherednik algebras
KW  - Hamiltonian reduction
KW  - Symplectic Lie algebra
L1  - https://knowledge.uchicago.edu/record/6424/files/Goyal_uchicago_0330D_16826.pdf
L2  - https://knowledge.uchicago.edu/record/6424/files/Goyal_uchicago_0330D_16826.pdf
L4  - https://knowledge.uchicago.edu/record/6424/files/Goyal_uchicago_0330D_16826.pdf
LA  - en
LK  - https://knowledge.uchicago.edu/record/6424/files/Goyal_uchicago_0330D_16826.pdf
N2  - Losev introduced the scheme X of almost commuting elements (i.e., elements commuting upto a rank one element) of g = sp(V) for a symplectic vector space V and discussed its algebro-geometric properties. We construct a Lagrangian subscheme Xnil of X and show that it is a complete intersection of dimension dim(g) + 1/2 dim(V ) and compute its irreducible components.

We study the quantum Hamiltonian reduction of the algebra D(g) of differential operators on the Lie algebra g tensored with the Weyl algebra with respect to the action of the symplectic group, and show that it is isomorphic to the spherical subalgebra of a certain rational Cherednik algebra of Type C. We contruct a category Cc of D-modules whose characteristic variety is contained in Xnil and construct an exact functor from this category to the category O of the above rational Cherednik algebra. Simple objects of the category Cc are mirabolic analogs of Lusztig's character sheaves.

We also define and study a group-theoretic version of Losev’s almost commuting scheme as well as the above quantum Hamiltonian reduction problem.
PB  - The University of Chicago
PY  - 2023-06
T1  - Almost Commuting Scheme of Symplectic Matrices and Quantum Hamiltonian Reduction
TI  - Almost Commuting Scheme of Symplectic Matrices and Quantum Hamiltonian Reduction
UR  - https://knowledge.uchicago.edu/record/6424/files/Goyal_uchicago_0330D_16826.pdf
Y1  - 2023-06
ER  -