REPRESENTATION THEORY OF SL 2 OVER A P-ADIC FIELD: THE PRINCIPAL SERIES

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1 REPRESENTATION THEORY O SL 2 OVER A P-ADIC IELD: THE PRINCIPAL SERIES ALEXANDER J. MUNK Abstract. A result concerning the irreducibilit reducibilit of the principal series of representations of SL 2 over a p-adic field is presented see Theorem 4.6. An overview of the structure of p-adic fields precedes the demonstration, as does the introduction of certain special functions on these fields. Contents. Introduction 2. p-adic ields 3. Special unctions on p-adic ields 3 4. The Principal Series of Representations of SL Acknowledgements 6. References. Introduction Let K be a locall compact Hausdorff nondiscrete topological field. The principal series of representations of SL 2 K is a collection of continuous unitar representations of SL 2 K on L 2 K. Its properties are well understood. We give here a brief analsis of the irreducibilit reducibilit of this series in the special case where K is a p-adic field p is odd. The computations techniques emploed follow [], [2], [3]. Section 2 contains necessar results concerning p-adic fields. The account is terse proofs are not provided. Readers desirous of further information should consult [3]. Section 3 treats certain special functions on p-adic fields. Proofs can be found in [3]. Section 4 introduces the principal series of representations of SL 2 over a p-adic field. A convenient unitaril equivalent representation is thoroughl studied. The paper concludes with the main result: Theorem p-adic ields Let p be an odd prime a finite algebraic extension of Q p. Denote the additive multiplicative groups of b +, respectivel. Let dx be a fixed Haar measure on +. Date: August 22, 28.

2 2 ALEXANDER J. MUNK Definition 2.. Define : R such that i is a non-archimedean norm on ; ii or an a, d ax = a dx. Note 2.2. There exists exactl one such function. Observe that dx/ x is a Haar measure on. Give the topolog induced b this norm. Define subsets O p of b O = {x : x } p = {x : x < }. O is the maximal compact subring of ; p is the unique maximal ideal of O. Moreover, p is principal. Let τ be a generator for p. O/p is a field with q elements, where q is some power of p. It can be shown that τ = q for all a, a = q n, for some n Z. Let U = {x : x = } be the group of units in. U contains an element ɛ such that i ɛ has order q ii = 2 τ 2 ɛτ 2 ɛ 2. Definition 2.3. Let = +,, or U. A character of is a continuous homorphism ψ : T, where T is the group of complex numbers with norm one. Denote the set characters of b ˆ. Define the following sets: p n = { x : x q n}, If χ ˆ, there exists s R with χ Û such that χ n Z U n = { x U : x q n} = + p n, n. π ln q < s π ln q x = x is χ u for all x, where x = q n xq n = u. or an nontrivial χ Û, there exists l such that χ is trivial on U l nontrivial on U l. or an nontrivial ψ ˆ +, there is an m Z such that ψ is trivial on p m nontrivial on p m. The following definitions are thus sensible. Definition 2.4. Let χ ˆ. χ is said to be unramified if χ is the trivial character, ramified of degree l otherwise, where χ l are as above. Definition 2.5. Let ψ ˆ + be nontrivial. Then p m is said to be the conductor of ψ, where m is as above.

3 REPRESENTATION THEORY O SL 2 OVER A P-ADIC IELD: THE PRINCIPAL SERIES 3 The three characters of degree two in ˆ figure prominentl below. them b sgn ɛ, sgn τ, sgn ɛτ, where { if x sgn ɛ x = 2 ɛ 2 otherwise { if x sgn θ x = 2 θ 2 otherwise for θ = τ or ɛτ. ix ψ ˆ + with conductor O. or all u, define ψ u ˆ + b for all x. ψ u x = ψ ux Denote Definition 2.6. Let f L. The ourier transform of f, f = ˆf, is defined b ˆf u = f x ψ u x dx for all u. restricted to L L 2 extends to an isometr of L 2. Denote this extension b as well. Without loss of generalit, assume that dx is normalized so that ˆf x = f x for all f L 2 x. Definition 2.7. The Schwarz-Bruhat space of, S, is the set of all complex-valued, compactl supported, locall constant functions on. Theorem 2.8. S is dense in L p, for p <. Theorem 2.9. The map defined b ϕ ˆϕ for all ϕ S is a bijection of S onto itself. 3. Special unctions on p-adic ields Note 3.. The special functions below are vital to section 4. This section characterizes them more full than required there, as the are also of independent interest. Definition 3.2. Let f : C be locall integrable, except possibl at. or all n, [f] n : C is defined b { f x if q [f] n x = n x q n otherwise. If the limit in exists, define the principal value integral of f b P.V. f x dx = lim [f] n n x dx. Theorem 3.3. Let f L 2. Suppose P.V. f x ψ u x dx exists for almost all u. Then ˆf u = P.V. for almost all u. f x ψ u x dx

4 4 ALEXANDER J. MUNK Note 3.4. This is part of a result known as Plancherel s theorem. Definition 3.5. Let χ be a nontrivial character of. Then is defined as follows: i if χ is ramified, Γ χ = Γ χ s = Γ χ s Γ χ = Γ χ s = P.V. ψ x χ x dx x. ii if χ is unramified R s >, Γ χ = Γ s = P.V. ψ x χ x dx x. iii if χ is unramified R s, Γ χ = Γ s is given b the analtic continuation of. Note 3.6. The above is well-defined. It is known as the gamma function. See [3]. Definition 3.7. Define q R b q + q =. Theorem 3.8. Let χ = χ s be a nontrivial multiplicative character on. i If χ is ramified of degree h, where Note that Γ χ = Γ χ s = C χ q hs 2, C χ = Γ χ /2. C χ = C χ C χ = χ. ii If χ is unramified, Γ χ = Γ s = qs q s. Γ s has a simple pole at s = with residue q ln q. / Γ s has a simple pole at s = with residue q ln q. The onl singularit of Γ s occurs at s = ; the onl zero occurs at s =.

5 REPRESENTATION THEORY O SL 2 OVER A P-ADIC IELD: THE PRINCIPAL SERIES 5 iii If χ x x, then Γ χ s = χ Γ χ s Γ χ s Γ χ s = χ. Hence, Γ χ s Γ χ s =. Definition 3.9. or χ ˆ u, v, define the Bessel function J χ u, v as follows: J χ u, v = P.V. ψ ux + v χ x x dx. x Note 3.. The Bessel function is well-defined. See [3]. Lemma 3.. Let u, v. Then i J χ u, v = J χ v, u. ii χ u J χ u, v = χ v J χ v, u. iii J χ u, v = J χ u, v = χ J χ u, v. iv If χ = resp., then J χ u, u is real-valued resp. pure imaginarvalued. Definition 3.2. Let k Z >, χ ˆ, v. Then v χ k, v = ψ x ψ χ x x dx. x =q x k Lemma 3.3. Suppose that v = q m k < m. i If χ is unramified, then χ k, v if onl if m is even k = m/2. ii If χ is ramified of degree h, then χ k, v if onl if one of the following holds: a m is even, m h, k = m/2 b m < 2h < 2m k = h or k = m h. Theorem 3.4. If χ ˆ is unramified, χ, u, v, then χ v Γ χ + χ u Γ χ uv q J χ u, v = χ u m χ 2, uv uv = q m, m >, m even uv = q m, m >, m odd. If χ, then the first case becomes J u, v = m + q 2 q = q [ ] ln uv + 2 ln q q for uv = q m q. The other cases remain valid as stated. Theorem 3.5. If χ ˆ is ramified of degree h, u, v, then χ v Γ χ + χ u Γ χ uv q h χ J χ u, v = u [ χ h, uv + χ m h, uv] uv = q m, h < m < 2h χ u m χ 2, uv uv = q m, m 2h, m even uv = q m, m > 2h, m odd..

6 6 ALEXANDER J. MUNK 4. The Principal Series of Representations of SL 2 Theorem 4.. Let χ ˆ, f L 2, x, α β SL γ δ 2. If [ α π χ γ β δ ] f x = χ βx + δ βx + δ f αx + γ βx + δ for all f L 2 almost all x, then π χ is a continuous unitar representation of SL 2 on L 2. Proof. See [] or [3]. Definition 4.2. The collection of representations {π χ : χ ˆ } is called the principal series of representations of SL 2. Definition 4.3. or all χ ˆ g SL 2, set g = π χ g. Note 4.4. Let χ ˆ. is a representation of SL 2 on L 2. It is unitaril equivalent to π χ more tractable, computationall. Lemma 4.5. Let γ, ϕ S, χ ˆ. Then for all x, ϕ x = ψ γx ϕ x. Proof. Suppose ϕ = ˆf. or all x, Hence, for all x. π χ ϕ x = ϕ x + γ. ϕ x =π χ = = π χ =f x + γ. ϕ x ϕ ψ x d f + γ ψ x d =ψ x γ ˆf x =ψ γx ϕ x Lemma 4.6. Let α, ϕ S, χ ˆ. Then for all x, α ϕ x = χ α α ϕ α 2 x. α

7 REPRESENTATION THEORY O SL 2 OVER A P-ADIC IELD: THE PRINCIPAL SERIES 7 Proof. Suppose ϕ = ˆf. or all x, Hence, for all x. π χ α α α α ϕ x =χ α α ϕ α 2 x =χ α α f α 2 x. α ϕ x =π χ ϕ x α α = π χ α =χ α α f α 2 ψ x d =χ α α f ψ x α 2 d =χ α α ϕ α 2 x ϕ ψ x d Lemma 4.7. Suppose L : L 2 L 2 is a bounded linear operator such that for all γ, α, χ ˆ, L = ˆπ χ L L α α = α α Then there exists m L such that for all f L 2, Lf x = m x f x L. for almost all x m is almost everwhere constant on the cosets of 2 in. Proof. Lemma 4.5 implies the first half. The required details are intricate. See [] or [3]. To obtain the second half, take the first as given note that for all α, ϕ S almost all x, [ L α α ϕ ] x = m x χ α α ϕ α 2 x, while [ α α b Lemma 4.6. ] Lϕ x = χ α α m α 2 x ϕ α 2 x Note 4.8. Examination of the algebra of bounded linear operators on L 2 commuting with, for χ ˆ not of order 2, ultimatel ields the proof of the irreducibilit of. Recall the following from [4]:

8 8 ALEXANDER J. MUNK Theorem 4.9. Let χ ˆ, x, ϕ S. The principal value integral x P.V. ψ χ ϕ d exists. Moreover, P.V. Note 4.. f S implies that ψ x χ ˆϕ d = ϕ u J χ u, x du. f u J χ u, x du is absolutel convergent for all χ ˆ x. See [4]. Lemma 4.. Let ϕ S χ ˆ. or almost all x, ϕ x = ϕ u J χ x, u du Proof. Suppose ϕ = ˆf. B Theorems , x P.V. χ f ψ d exists for all x. Clearl, x P.V. χ f ψ d = P.V. Theorem 3.3 implies π χ χ f ψ x d. f x = P.V. χ f ψ x d for almost all x. Theorem 4.9 Lemma 3. give x P.V. χ f ψ d x =P.V. χ ˆf ψ d = ˆf u J χ u, x du = ˆf u J χ x, u du for all x. Hence, for almost all x. ϕ x = ϕ u J χ x, u du Lemma 4.2. Let χ ˆ have order different from 2. If C C 2 are distinct cosets of 2 in, then there exist sets A C A 2 C 2 of positive measure such that x A u A 2 implies J χ x, u.

9 REPRESENTATION THEORY O SL 2 OVER A P-ADIC IELD: THE PRINCIPAL SERIES 9 Proof. Take C = 2 C 2 = ɛ 2. The other cases are similar., 2q Suppose χ. Let N = max q. Set { } A = : q N 2 2 A 2 = { : q N 2 Theorem 3.4 gives the result. Suppose χ is unramified. Set } ɛ 2. A = { : q} 2 A 2 = { : q} ɛ 2. Lemma 3.3 Theorem 3.4 finish this case. Suppose χ is ramified of degree h. Set A = { : q h} 2 A 2 = { : q h} ɛ 2. Lemma 3.3 Theorem 3.5 complete the proof. Note 4.3. The result above requires the hpothesis concerning the order of χ. Take C = 2, C 2 = τ 2, χ = sgn ɛ to observe this. The computation is eas. Theorem 4.4. Let χ ˆ. Suppose that χ is not of order 2. Then is irreducible. Proof. Suppose L : L 2 L 2 is a bounded linear operator such that L commutes with α, ˆπ χ, ˆπ α χ for all γ α. B Lemma 4.7, there exists m L such that for all f L 2, Lf x = m x f x for almost all x m is almost everwhere constant on the cosets of 2 in. It suffices to show that m is almost everwhere constant on. or ϕ S almost all x, ϕ u m u J χ x, u du = ϕ u m x J χ x, u du b Lemma 4.. Hence, m u J χ x, u = m x J χ x, u for almost all x, u. The result follows b Lemma 4.2. Theorem 4.5. Let χ ˆ have order 2. Then is reducible.

10 ALEXANDER J. MUNK Proof. The characters of order 2 on are sgn ɛ, sgn τ, sgn ɛτ. sgn τ sgn ɛτ are ramified of degree. sgn ɛ is unramified. Let θ = ɛ, τ, or ɛτ. Let ϕ S. Theorems Lemma 4. give ϕ x = = + ϕ u J χ x, u du xu q ϕ u Γ sgn θ [sgn θ u + sgn θ x] du m>, m even m ϕ u sgn θ x sgn θ xu =q 2, xu du m for almost all x. Suppose that ϕ is supported on ker sgn θ. Then ϕ u Γ sgn θ [sgn θ u + sgn θ x] du where xu q =Γ sgn θ [ + sgn θ x] A ϕ u du A = ker sgn θ {u : xu q}. If x / ker sgn θ, vanishes. Let n >, n even. ix u ker sgn θ suppose xu = q n, where x. Then xu sgn θ d ψ ψ =q n/2 = Hence, if x / ker sgn θ, =q n/2 ψ xu xu ψ sgn θ =sgn θ x ψ ψ =q n/2 =q n/2 ψ ψ xu d sgn θ d. xu sgn θ d =. But n ϕ u sgn θ x sgn θ xu =q 2, xu du n = ϕ u sgn θ x ψ ψ xu =q n =q n/2 m>, m even xu sgn θ d du. Hence, x / ker sgn θ, implies that m ϕ u sgn θ x sgn θ xu =q 2, xu du =, m so L 2 ker sgn θ

11 REPRESENTATION THEORY O SL 2 OVER A P-ADIC IELD: THE PRINCIPAL SERIES is invariant under the action of b Theorem 2.8. Theorem 2.8 Lemmas indicate that α ˆπ χ α also fix this space. Since matrices of the form α,, α where γ α, generate SL 2, the result follows. Theorem 4.6. Let χ ˆ. If χ has order 2, then π χ is reducible. Otherwise, π χ is irreducible. Proof. is unitaril equivalent to π χ. Cite Theorems Acknowledgements We are indebted to Professor Sall. Without his suggestion of the project, careful supervision unceasing encouragement, this endeavor would not have been possible. 6. References [] Boarchenko, Dmitri. Irreducibilit of the Principal Series Representations of SL 2 over a Local ield. Unpublished Notes. 28, -9. [2] Gel f, I.M. Graev, M.I. Representations of a Group of Matrices of the Second Order with Elements from a Locall Compact ield, Special unctions on Locall Compact ields. Uspehi Matematichesm Nauk , 29-. [3] Sall, Jr. P.J. An Introduction to p-adic ields, Harmonic Analsis the Representation Theor of SL 2. Letters in Mathematical Phsics , -47. [4] Sall, Jr. P.J. Taibleson, M.H. Special unctions on Locall Compact ields. Acta. Math , ,