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Geometric construction of crystal bases. (English) Zbl 0901.17006
Let $${\mathfrak g}$$ denote a complex semisimple Lie algebra. G. Lusztig has constructed a basis (called the canonical basis) of $$U_q^-({\mathfrak g})$$ by considering perverse sheaves on quiver varieties [J. Am. Math. Soc. 3, 447-498 (1990; Zbl 0703.17008)]. In this paper the authors give an example of such a simple perverse sheaf whose singular support is not irreducible. The example is for type $$A_5$$. It gives a negative answer to a problem posed by G. Lusztig in [J. Am. Math. Soc. 4, 365-421 (1991; Zbl 0738.17011)].
The authors proceed to establish a connection to a conjecture by D. Kazhdan and G. Lusztig [Adv. Math. 38, 222-228 (1980; Zbl 0458.20035)] on the characteristic variety of the regular holonomic $${\mathfrak D}$$-module associated to a highest weight module for $${\mathfrak g}$$. This conjecture says that for type $$A$$ such varieties are always irreducible. Here the authors prove this conjecture for type $$A_n$$, $$n<7$$, and give an example which shows that it fails for type $$A_7$$.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 20G05 Representation theory for linear algebraic groups
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##### References:
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