Differential operators on quantized flag manifolds at roots of unity. II. (English) Zbl 1311.14047

Tanisaki, Toshiyuki

Nagoya Math. J. 214, 1-52 (2014).

Let \(G\) be a connected, simply connected simple algebraic group over \(\mathbb{C}\) and \(\ell\) an odd integer greater than the Coxeter number of \(G\). Fixing a primite \(\ell\)th root of unity \(\zeta\), the De Concini-Kac quantized enveloping algebra of \(\mathrm{Lie} \;G\) is \(U_{\zeta}\). If \(\mathcal{B}_{\zeta}\) is the quantized flag manifold (a non-commutative space), it is possible to define the sheaf \(\mathcal{D}_{\mathcal{B}_{\zeta},t}\) of twisted differential operators on \(\mathcal{B}_{\zeta}\); the twist \(t\) belonging to \(H\), a maximal torus of \(G\). The Frobenius morphism \(\mathrm{Fr} : \mathcal{B}_{\zeta} \rightarrow \mathcal{B}\) is, in an appropriate sense, a finite morphism, where \(\mathcal{B}\) is the usual flag manifold. Then \(\mathrm{Fr}_* \mathcal{D}_{\mathcal{B}_{\zeta},t}\) is a genuine sheaf of non-commutative algebras on \(\mathcal{B}\). The category of coherent \(\mathrm{Fr}_* \mathcal{D}_{\mathcal{B}_{\zeta},t}\)-modules is denoted \(\mathrm{Mod}_{\mathrm{coh}}(\mathrm{Fr}_* \mathcal{D}_{\mathcal{B}_{\zeta},t})\). The Harish-Chandra center \(Z_{\zeta}\) of \(U_{\zeta}\) can be identified with \(\mathbb{C}[H/W]\). If \(t \in H\), we denote by \(U_{\zeta,t}\) the corresponding central quotient of \(U_{\zeta}\) and by \(\mathrm{Mod}_f (U_{\zeta,t})\) the category of finitely generated \(U_{\zeta,t}\)-modules.
Assume that \(t\) is regular. In this article the author states a conjectural derived equivalence between the bounded derived category of \(\mathrm{Mod}_f (U_{\zeta,t})\) and the bounded derived category of \(\mathrm{Mod}_{\mathrm{coh}}(\mathrm{Fr}_* \mathcal{D}_{\mathcal{B}_{\zeta},t})\). The derived equivalence should be given by the global sections functor \(R \Gamma(\mathcal{B}, - )\). The conjecture is motivated by the analogy between \(U_{\zeta}\) and the enveloping algebra of \(\mathrm{Lie} \;G\) in positive characteristic and the corresponding derived equivalence established in [R. Bezrukavnikov et al., Ann. Math. (2) 167, No. 3, 945–991 (2008; Zbl 1220.17009)]. Using the arguments of [loc. cit.], the author shows that the conjecture is true provided \(R \Gamma (\mathcal{B}, \mathrm{Fr}_* \mathcal{D}_{\mathcal{B}_{\zeta},t}) \simeq U_{\zeta,t}\). The author states a more general form of the conjecture, where the central quotients defined by \(t\) are replaced by generalized central characters.
In the final part of the article, the author states a different conjecture involving a certain induced \(U_{\zeta,t}\)-module. Remarkably, it is shown that this conjecture is equivalent to the above stated conjecture.
For Part I and III, see [the author, Adv. Math. 230, No. 4–6, 2235–2294 (2012; Zbl 1264.14067); Adv. Math. 392, Article ID 107990, 51 p. (2021; Zbl 1483.14086)].

Reviewer: Gwyn Bellamy (Glasgow)

MSC:

14M15	Grassmannians, Schubert varieties, flag manifolds
20G05	Representation theory for linear algebraic groups
17B37	Quantum groups (quantized enveloping algebras) and related deformations
14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14A22	Noncommutative algebraic geometry
32C38	Sheaves of differential operators and their modules, \(D\)-modules

Citations:

Zbl 1220.17009; Zbl 1264.14067; Zbl 1483.14086

References:

[1]	M. Artin and J. J. Zhang, Noncommutative projective schemes , Adv. Math. 109 (1994), 228-287. · Zbl 0833.14002
[2]	E. Backelin and K. Kremnizer, Quantum flag varieties, equivariant quantum \(\mathcal{D}\)-modules, and localization of quantum groups , Adv. Math. 203 (2006), 408-429. · Zbl 1165.17304
[3]	E. Backelin, K. Kremnizer, Localization for quantum groups at a root of unity , J. Amer. Math. Soc. 21 (2008), 1001-1018. · Zbl 1248.14003
[4]	E. Backelin, K. Kremnizer, Global quantum differential operators on quantum flag manifolds, theorems of Duflo and Kostant , preprint, [math.RT]. 1106.4343
[5]	R. Bezrukavnikov, I. Mirković, and D. Rumynin, Singular localization and intertwining functors for reductive Lie algebras in prime characteristic , Nagoya Math. J. 184 (2006), 1-55. · Zbl 1125.17006
[6]	R. Bezrukavnikov, I. Mirković, D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic , Ann. of Math. (2) 167 (2008), 945-991. · Zbl 1220.17009
[7]	K. A. Brown and K. R. Goodearl, Homological aspects of Noetherian PI Hopf algebras of irreducible modules and maximal dimension , J. Algebra 198 , (1997), 240-265. · Zbl 0892.16022
[8]	C. De Concini and V. G. Kac, “Representations of quantum groups at roots of 1” in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989) , Progr. Math. 92 , Birkhäuser, Boston, 1990, 471-506. · Zbl 0738.17008
[9]	C. De Concini and C. Procesi, “Quantum groups” in \(D\)-modules, Representation Theory and Quantum Groups (Venice, 1992) , Lecture Notes in Math. 1565 , Springer, Berlin, 1993, 31-140. · Zbl 0795.17005
[10]	F. Gavarini, Quantization of Poisson groups , Pacific J. Math. 186 (1998), 217-266. · Zbl 0921.17004
[11]	A. Joseph, Quantum Groups and Their Primitive Ideals , Ergeb. Math. Grenzgeb. (3) 29 , Springer, Berlin, 1995. · Zbl 0808.17004
[12]	A. Joseph and G. Letzter, Separation of variables for quantized enveloping algebras , Amer. J. Math. 116 (1994), 127-177. · Zbl 0811.17007
[13]	G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras , Adv. Math. 70 (1988), 237-249. · Zbl 0651.17007
[14]	G. Lusztig, Quantum groups at roots of \(1\) , Geom. Dedicata 35 (1990), 89-113. · Zbl 0714.17013
[15]	G. Lusztig, Introduction to Quantum Groups , Progr. Math. 110 , Birkhäuser, Boston, 1993. · Zbl 0788.17010
[16]	N. Popescu, Abelian Categories with Applications to Rings and Modules , London Math. Soc. Monogr. Ser. 3 , Academic Press, London, 1973. · Zbl 0271.18006
[17]	R. Steinberg, On a theorem of Pittie , Topology 14 (1975), 173-177. · Zbl 0318.22010
[18]	T. Tanisaki, “Killing forms, Harish-Chandra isomorphisms, and universal \(R\)-matrices for quantum algebras” in Infinite Analysis, Part A, B (Kyoto, 1991) , Adv. Ser. Math. Phys. 16 , World Sci. Publ., River Edge, New Jersey, 1992, 941-961. · Zbl 0870.17007
[19]	T. Tanisaki, The Beilinson-Bernstein correspondence for quantized enveloping algebras , Math. Z. 250 (2005), 299-361. · Zbl 1082.17010
[20]	T. Tanisaki, Differential operators on quantized flag manifolds at roots of unity , Adv. Math. 230 (2012), 2235-2294. · Zbl 1264.14067
[21]	T. Tanisaki, Manin triples and differential operators on quantum groups , Tokyo J. Math., 36 (2013), 49-83. · Zbl 1301.17015