By Michel Raynaud

ISBN-10: 3540052836

ISBN-13: 9783540052838

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Finally, √ n √ n 2 √ n 3 c(α) = 1 + + 2( ) + 4( ) + ··· , so that Im c(α) = (n−1)/2 +4 (3n−1)/2 + ··· ≡ (n−1)/2 = Im (α) (mod 1 + ℘). The next two technical lemmas will come in handy in working out the arithmetic of the character formulas. 6. Write ψ0 for the quadratic character of C (so that ψ02 = 1, but ψ0 = 1). We shall also use ψ0 later for quadratic characters on related groups. 3]). For λ ∈ C , ⎧ ⎨− sgn (−1), ψ0 (λ) = ⎩ sgn (λ + λ−1 + 2), λ = −1 otherwise. Proof. Clearly, ψ0 (1) = 1 = sgn (1 + 1−1 + 2).

1. Note that, in this setting, H/T is a singleton. Proof. 7. 8. 1]). Suppose x ∈ B opt , r ≥ 0, and let χ be an irreducible representation of Gx,r , trivial on Gx,r+ . 1) contains no nilpotent elements. Two minimal K-types (Gx,r , χ) and (Gy,s , ξ) are associate if r = s and (a) r = 0, x = y, and χ is equivalent to ξ; or (b) r > 0, and the G-orbit of the coset that realizes χ intersects the coset that realizes ξ. For arbitrary reductive, p-adic groups, one must call depth-zero K-types associate even under some circumstances when x = y; but working with G = SL2 , and restricting to x ∈ Bopt , avoids this complication.

3. 2 (denoted by ρ(π) in [33]) is called the depth d(π) of π. 4. If the representation π of G contains an unreﬁned minimal Ktype of the form (GxR ,0 , χR ), then π ◦ Int ( 10 0 ) contains the unreﬁned minimal K-type (GxL ,0 , χL ), where χL = χR ◦ Int ( 10 0 ). 38 20 ADLER, DEBACKER, SALLY, AND SPICE In order to classify the representations of G, we start by listing the unreﬁned minimal K-types for G, and check which items on the list are associate. 9. 1. Cuspidal representations of SL2 (f). Write G = SL2/f .

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