Papalexiou Nikolaos

Title:”On the prime spectrum of the enveloping algebra and characteristic varieties”.

ABSTRACT: Let 𝔤 be a complex semisimple Lie algebra, 𝔥 be a Cartan subalgebra of 𝔤 and Spec(U(𝔤)) the set of prime ideals of U(𝔤). In order to describe Spec(U(𝔤)), W. Soergel in [3] attaches to every irreducible subset Ω of 𝔥^{⋆}, a prime ideal I_{Ω} in Spec(U(𝔤)), such that I_{Ω} = ⋂_{λ∈Ω} I_{λ}, where I_{λ} is the annihilator of the simple highest weight module L(λ). In [2], A. Joseph attaches to a two-sided ideal I of U(𝔤) a subvariety V(I) of 𝔥^{⋆}, called characteristic variety of I. Using these varieties we can find the order relations between prime ideals. Our aim is the calculation of the characteristic variety of I_{Ω}. Denote by R_{Ω} the integral root system corresponding to Ω, W_{Ω} the associated Weyl group and D_{Ω} the subset of w ∈ W such that w(R_{Ω} ∩ R^{+}) ⊂ R^{+}. Generalizing the notion of regular dominant for an irreducible Ω ⊂ 𝔥^{⋆} we want to prove that:

Conjecture 1: Let Ω be a regular dominant irreducible subset of 𝔥^{⋆} and I_{Ω} the prime ideal defined by Ω. Then the characteristic variety of I_{Ω} is given by:

V(I_{y.Ω}) = D_{Ω}\overline{C_{Ω}(y)}.Ω,

where \overline{C_{Ω}(y)} = \{w ∈ W_{Ω} | I_{w.Ω} ⊃ I_{y.Ω}\} is the left cone of W_{Ω}.

This conjecture implies the conjecture posed by W. Soergel in [3]:

Conjecture 2. [3]: For any irreducible subset Ω in 𝔥^{⋆} we have I_{Ω} ⊂ I_{λ} if and only if there exists w ∈ W such that I_{Ω} ⊂ I_{w.Ω} and λ ∈ w.Ω.

References

[1] : Borho W. and Joseph A., Sheets and Topology of Primitive Spectra for semisimple Lie algebras, to appear.

[2] : Joseph A., A characteristic variety for the primitive spectrum of a semisimple Lie algebra, Lecture Notes in Mathematics, 587, Springer, Berlin 1977, p. 102-118.

[3] : Soergel W., The prime spectrum of the enveloping algebra of a reductive Lie algebra, Mathematische Zeitschrift, 204, p. 559-581, 1990.