By A. I. Kostrikin, I. R. Shafarevich

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**Additional info for Algebra II - Noncommunicative Rings, Identities**

**Example text**

Deﬁne an element of Gder (R)0 / δθ δθ δθ × Sder (R) ∩ Sder (R)0 to be equivalent to (g Sder (R), s ) ∈ Gder (R)0 / δθ δθ −1 × Sder (R) if it is of the form (g wSder (R), w s δθ(w)) for some w ∈ δθ NGder (R)0 (Sder (R)0 δθ). Apparently, equivalent elements have the same image unδθ (R)0 δθ)| elements in any equivalence class. der Ψ, and there are |Ω(Gder (R)0 , Sder δθ Conversely, suppose that Ψ(gSder (R), s) = Ψ(g Sder (R), s ). We are to show δθ that (gSder (R), s) is equivalent to (g Sder (R), s ).

Thus, for each representation τ ∈ XLirr (Λ1 ) we obtain from (37) an operator τ ⊗ SΛ1 (δθ) which intertwines 1 with 1δθ . We describe the representation τ by ﬁrst recalling that Λ1 = δθ · Λ1 (Lemma 5) and setting τ0 (exp(X)) = e(iΛ1 +ρ1 )(X) , X ∈ sder (41) This deﬁnes a character in L0 (Λ1 ). We may extend it to a character of L(Λ1 ) by recalling (36), choosing an th root of unity ζ, and setting τ0 (δθ) = ζ e(iΛ1 +ρ1 )(log(s0 )/ ) . (42) This produces a character τ0 of L(Λ1 ) which depends on a choice of an th root of unity.

R } ⊂ R(H 1 r ˆ that ϕH1 (WR ) lies in the centralizer in H1 of ∩j=1 ker βj . It follows that ϕH1 (WR ) permutes the positive roots not generated by {β1 , . . , βr }. Let β ∈ X ∗ (TˆH1 ) be the sum of the positive roots not generated by {β1 , . . , βr }. We may identify ˆ 1 , TˆH ) with the set of positive roots in R(H, ˆ TH ) the set of positive roots in R(H 1 with respect to the Borel subgroup BH . The fact that ξH1 extends the embedˆ → H ˆ 1 implies that ξ −1 ◦ ϕH (WR ) still lies in the corresponding cending H 1 H1 ˆ The map ξ allows us to identify R(H, ˆ TH ) with a subsystem of tralizer in H.

### Algebra II - Noncommunicative Rings, Identities by A. I. Kostrikin, I. R. Shafarevich

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