aa r X i v : . [ m a t h . QA ] J u l GENERIC SIMPLICITY OF QUANTUM HAMILTONIANREDUCTIONS
AKAKI TIKARADZE
Abstract.
Let a reductive group G act on a smooth affine complex algebraicvariety X. Let g be the Lie algebra of G and µ : T ∗ ( X ) → g be the momentmap. If the moment map is flat, and for a generic character χ : g → C , theaction of G on µ − ( χ ) is free, then we show that for very generic characters χ the corresponding quantum Hamiltonian reduction of the ring of differentialoperators D ( X ) is simple. Let a reductive algebraic group G act on a smooth affine algebraic variety X over C . Let g be the Lie algebra of G. Let µ : T ∗ ( X ) → g ∗ be the correspondingmoment map. We will assume that this map is flat, and for generic G -invariantcharacter χ ∈ g ∗ the action of G on µ − ( χ ) is free.Given a G -invariant character χ ∈ g ∗ , denote by U χ ( G, X ) the quantum Hamil-tonian reduction of D ( X ) with respect to χ . So, U χ ( G, X ) = ( D ( X ) /D ( X ) g χ ) G , where g χ = { g − χ ( g ) ∈ D ( X ) , g ∈ g } . The usual filtration on D ( X ) by the orderof differential operators induces the corresponding filtration on U χ ( G, X ) . Thenit follows from the flatness of the moment map thatgr U χ ( G, X ) = O ( µ − (0) //G ) . In what follows by a very generic subset we mean a complement of a union ofcountably many proper closed Zariski subsets. Under these assumptions we havethe following result.
Theorem 0.1.
For very generic values of a G -invariant character χ ∈ g ∗ , thecorresponding quantum Hamiltonian reduction U χ ( G, X ) is simple. Moreover, if f ∈ g / [ g , g ] is so that G acts freely on µ − ( χ ) whenever χ ( f ) = 0 , then U χ ( G, X ) is simple for all χ such that χ ( f ) / ∈ Q . The proof is be based on the reduction modulo p n technique for a large prime p. At first, we recall that given a ring R such that p is not a zero divisor, then thecenter of its reduction modulo p, R p = R/pR acquires a natural Poisson bracket,to be referred to as the reduction modulo p Poisson bracket, defined as follows.Given central elements x, y ∈ Z ( R p ) , let x ′ , y ′ ∈ R be their lifts. Then { x, y } = ( 1 p [ x ′ , y ′ ]) mod p ∈ Z ( R p ) . We use the following result [[T], Corollary 8].
Lemma 0.1.
Let k be a perfect field of characteristic p. Let A be a p -adicallycomplete topologically free W ( k ) -algebra, such that A = A/pA is an Azumayaalgebra over its center Z . Assume that
Spec( Z ) is a smooth symplectic k -varietyunder the reduction modulo p Poisson bracket. Then A [ p − ] is topologically sim-ple. Next we need to recall some results and notations associated with quantumHamiltonian reduction of the ring of crystalline differential operators in charac-teristic p from [BFG].Let X be a smooth affine variety over an algebraically closed field k of char-acteristic p , and G be a reductive algebraic group over k with the Lie algebra g . Denote by D ( X ) the ring of crystalline differential operators on X. As be-fore, we have the moment map µ : T ∗ ( X ) → g ∗ and the algebra homomorphism U ( g ) → D ( X ) . Now recall that the p -center of U ( g ), denoted by Z p ( g ) , is gener-ated by g p − g [ p ] , g ∈ g . We get an isomorphism i : Sym( g ) (1) → Z p ( g ) . On the other hand, the center of D ( X ) is generated by O ( X ) p and ξ p − ξ [ p ] , ξ ∈ T X and this leads to an isomorphism O ( T ∗ ( X )) (1) → Z ( D ( X )) . We have η ′ : Z p ( g ) → Z ( D ( X )) and the corresponding homomorphism η : Sym( g ) (1) → O ( T ∗ ( X )) (1) . Given χ ∈ g ∗ , then χ [1] ∈ g ∗ is defined as follows: χ [1] ( g ) = χ ( g ) p − χ ( g [ p ] ) , g ∈ g . Using the above homomorphisms it follows that the center of U χ ( G, X ) contains O ( µ − ( χ [1] ) //G ) . In this setting the following holds.
Lemma 0.2. [BFG]
Let χ ∈ ( g ∗ ) G be a character such that G acts freely of µ − ( χ [1] ) . Then U χ ( G, X ) is an Azumaya algebra over µ − ( χ [1] ) //G. We need the following criterion of simplicity of certain filtered quantizations.
Lemma 0.3.
Let S ⊂ C be a finitely generated ring, and let R be a filtered S -algebra, such that gr( R ) is a finitely generated commutative ring over S. Assumethat for all large enough primes p the algebra R p = R/pR is an Azumaya algebraover its center Z p , moreover Spec( Z p ) is a smooth symplectic variety over S p under the reduction modulo p Poisson bracket. Let F be the field of fractions of S. Then R F = R ⊗ S F is a simple ring.Proof. Let I be a nonzero two sided ideal of R such that R/I is not F -torsion.After localizing S further, we may assume using the generic flatness theoremthat gr( R/I ) and
R/I, R are free S -modules. Hence for p ≫ , ¯ I p (the p -adic ENERIC SIMPLICITY OF QUANTUM HAMILTONIAN REDUCTIONS 3 completion of I ) is a topologically free nontrivial two-sided ideal of ¯ R p (the p -adiccompletion of R ). Now Lemma 0.1 yields a contradiction. (cid:3) Next we state a result implying that taking quantum Hamiltonian reductionand reducing modulo a large prime commute. The statement and its proof werekindly provided by W. van der Kallen ( via mathoverflow.org.) Possible mistakesin the proof below are solely due to the author.
Theorem 0.2 (van der Kallen) . Let S be a commutative Noetherian ring of finitehomological dimension, let R be a commutative S -algebra flat over S. Let G be asplit reductive group over S acting on R. Then for all p ≫ and a base changeto a characteristic p field S → k , the map R G ⊗ S k → R G k k is surjective.Proof. At first, recall that there exists an integer n ≥ H i ( G, S [ n ]) = 0for all i [[FW], Theorem 33]. This implies H i ( G, S [ n ] ⊗ S N ) = 0 for any S -module N with the trivial G -action (since S has a finite global dimension). Let D, N berespectively the image and kernel of the map S [ n ] → k . As H ( G, S [ n ] ⊗ S N ) = 0 , we get that ( R ⊗ S S [ n ]) G → ( R ⊗ S D ) G is surjective. Now flatness of k over D yields that ( R ⊗ S D ) G ⊗ D k = R G k k . Therefore, we obtain the desired surjectivity R G ⊗ S k → R G k k (cid:3) Proof of Theorem 0.1.
Recall that 0 = f ∈ O (( g ∗ ) G ) has the property that forany χ ∈ ( g / [ g , g ]) ∗ such that f ( χ ) = 0 , the action of G on µ − ( χ ) is free. Let S ⊂ C be a large enough finitely generated subring over which X, f and theaction of G on X are defined.Let e , · · · , e l be a basis of g / [ g , g ] over S. Let W ⊂ ( g / [ g , g ]) ∗ be the set of all χ so that χ ( e i ) are algebraically independent over S. Clearly W is a very genericsubset. We will show that for any χ ∈ W, U χ ( G, X ) is simple.Put R = U χ ( G, X ) . We verify that R satisfies assumptions in Lemma 0.3.Indeed, it follows from Proposition 0.2, that for a base change S → k to an alge-braically closed field k of characteristic ≫ , R ⊗ S k is isomorphic to U ¯ χ ( G k , X k ) , where ¯ χ denotes the base change of χ. Then G k acts freely on µ − ( ¯ χ ) . So Lemma0.2 implies that U ¯ χ is an Azumaya algebra over a symplectic variety under thereduction modulo p Poisson bracket. So, conditions of Lemma 0.3 are met.Let f ∈ g / [ g , g ] be such that G acts freely on µ − ( χ ) when χ ( f ) = 0 . Wemay assume that S contains χ ( e i ) . Then it follows from the Chebotarev densitytheorem that there are arbitrarily large primes p and a base change S → k , sothat the image of χ ( f ) is nonzero. So, just as above we may conclude that thealgebra U χ ( G, X ) is simple. (cid:3)
We may apply the above result to certain filtered quantizations of quiver va-rieties as follows. Let Q be a quiver with n vertices, let α be a its positiveroot. Then G = Q GL α i / C ∗ acts on the space of α -dimensional representations AKAKI TIKARADZE
Rep ( Q, α ) giving rise to the moment map m α : T ∗ ( Rep ( Q, α )) → g ∗ . We willidentify ( g ∗ ) G with λ ∈ C n such that λ · α = 0 . From now on we assume that themoment map m α is flat. The set of such dimension vectors α was fully describedby Crawly-Boevey in [[CB] Theorem 1.1] Denote by A λ ( Q, α ) the correspondingquantum Hamiltonian reduction of the ring of differential operators D ( Rep ( Q, α ))with respect to the character λ. We have the following direct corollary of Theorem 0.1. Remark that strongerresults on generic simplicity follows from the works of Losev on quantizations ofquiver varieties (see for example [[L], Theorem 1.4.2].)
Theorem 0.3.
Let α be a positive root as above. Let λ · α = 0 be such that λ · β / ∈ Q for any positive root β < α. Then A λ ( Q, α ) is simple. References [BFG] R. Bezrukavnikov, M. Finkelberg, V. Ginzburg,
Cherednik algebras and Hilbert schemesin characteristic p , Represent. Theory 10 (2006), 254–298.[CB] W. Crawley-Boevey, Geometry of the moment map for representations of quivers , Com-positio Math. 126 (2001), 257–293.[L] I. Losev,
Completions of symplectic reflection algebras , Completions of symplectic reflectionalgebras. Selecta Math. (N.S.) 18 (2012), no. 1, 179–251.[T] A. Tikaradze,
Ideals in deformation quantizations over Z /p n Z , J. of Pure and AppiedAlgebra 221 (2017) 229–236.[FW] V. Franjou, W. van der Kallen, Power reductivity over an arbitrary base of change , Doc.Math. 2010, Extra vol. Andrei A. Suslin sixtieth birthday, 171–195. , E-mail address : [email protected]@gmail.com