REPRESENTATIONS OF REDUCTIVE p-ADIC GROUPS Under revision March 2009 Table of Contents 1. Introduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 2 2. Valuations and local fields :::::::::::::::::::::::::::::::::::::::::::::::::::: 5 3. Smooth representations of locally compact totally disconnected groups :::::::: 11 4. Haar measure, convolution, and characters of admissible representations:::::::: 21 5. Induced representations - general properties ::::::::::::::::::::::::::::::::::: 29 6. Parabolic induction and Jacquet modules ::::::::::::::::::::::::::::::::::::: 37 7. Supercuspidal representations and Jacquet's subrepresentation theorem :::::::: 50 8. Depth zero supercuspidal representations ::::::::::::::::::::::::::::::::::::: 63 9. Parahoric subgroups in symplectic groups ::::::::::::::::::::::::::::::::::::: 70 10. Cuspidal representations and the Deligne-Lusztig construction ::::::::::::::: 73 11. Tori over Fq and unramified tori in p-adic groups :::::::::::::::::::::::::::: 74 12. Depth zero supercuspidal L-packets :::::::::::::::::::::::::::::::::::::::::: 76 13. Hecke algebras and K-types for smooth representations :::::::::::::::::::::: 77 14. Discrete series representations ::::::::::::::::::::::::::::::::::::::::::::::: 86 15. Asymptotic behaviour of matrix coefficients of admissible representations ::::: 92 16. More about Jacquet modules :::::::::::::::::::::::::::::::::::::::::::::::: 100 17. Linear independence of characters ::::::::::::::::::::::::::::::::::::::::::: 102 18. Principal series representations :::::::::::::::::::::::::::::::::::::::::::::: 104 19. Intertwining maps ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 110 20. The Satake isomorphism ::::::::::::::::::::::::::::::::::::::::::::::::::::: 115 21. Macdonald's formula and spherical representations ::::::::::::::::::::::::::: 120 22. References :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 127 1 1. Introduction Let F be a field and n a positive integer. The set G = GLn(F ) of invertible matrices with entries in the field F forms a group with respect to matrix multiplication. This is an example of a reductive group. If F is a finite field, then G is a finite group. As (complex) representations of finite groups are discussed in earlier courses, some results concerning representations of G in the case when F is a finite field will be assumed. In this course, we will be particularly interested in the case where F is a p-adic field (p-adic fields will be described in section 2). But we can ask what are the representations of G for any field F . A representation of G consists of a (possibly infinite dimensional) complex vector space V and a homomorphism π from G to the set of invertible linear operators on V . Depending on the field F , we restrict our attention to representations satisfying \extra" topological conditions. We make a few remarks about the connection between representations of p-adic groups and automorphic forms. Suppose that F is a number field (a finite extension of the rational numbers). Then F has completions with respect to the various valuations on F . For example, if F is the rational numbers, the completion of F with respect to the standard valuation (absolute value) j · j1 is the real numbers. For each prime integer p, the completion of F with respect to the p-adic valuation j· jp is the p-adic numbers Qp (see Q section 2). The adelic ring A of Q is a certain subring of the direct product R× p Qp of all of these completions. Similarly we can form the adelic ring AF of a general number field F . Given a reductive group G which is defined over F , we can define the adelic group G(AF ) (e.g. GLn(AF )). An automorphic form on G(AF ) is a complex-valued function on G(AF ) satisfying certain conditions (not discussed here). Now suppose that V is a subspace of the vector space of complex-valued functions on G(AF ), having the property that for f 2 V and g 2 G(AF ), the function x 7! f(xg) also belongs to V . Then we have a natural representation π on V :(π(g)f)(x) = f(xg). If V happens to consist of automorphic forms, then π is said to be an automorphic representation of G(AF ). The group G(AF ) can be expressed as a restricted direct product of the groups G(Fv), as Fv runs over the various completions of F . Also, the automorphic representation π decomposes into a tensor product of representations πv, where πv is a representation of G(Fv). The fields Fv are called local fields, and the representations πv are called the local components of π. It turns out that for all but finitely many completions, Fv is a p-adic field. So most of the local components of automorphic representations are representations of reductive p-adic groups. 2 (The other components are also important - they are representations of real or complex reductive groups). Now we discuss harmonic analysis on topological groups. A topological group G is a group which is also a topological space and has the property that the map (x; y) 7! xy−1 from G × G to G is continuous. Assume that G is (the rational points of) a reductive group over a p-adic field: such a group is a locally compact topological group. There exists a measure µ on G which is G-invariant: µ(gS) = µ(Sg) = µ(S), for g 2 G, and S any measurable subset of G. This measure µ is unique up to positive scalar multiples and is called Haar measure on G. Let L2(G) denote the set of complex valued functions on G which are square integrable relative to Haar measure. The representation R of G defined by (R(y)f)(x) = f(xy), x; y 2 G, f 2 L2(G), is called the right regular representation. One of the main problems in harmonic analysis on G is to decompose the right regular representation, that is, to express it in terms of irreducible representations of G. Another major problem is to determine Plancherel measure. This involves character theory and Fourier transforms. Associated to an irreducible representation π of G (satisfying certain smoothness conditions) is a distribution Θπ, the character of π. This distribution is a linear functional on some vector space V of complex valued functions on G. In the case of P finite groups Θπ(f) = g2G trace(π(g)) f(g). In general, π(g) is an operator on an infinite dimensional vector space and so does not have a trace - but if we replace the finite sum with integration with respect to Haar measure, and move the trace outside the integral, and make enough assumptions on π and V, then we are taking the trace of an operator which has finite rank, so Θπ(f) is defined. Let Gb be the set of irreducible unitary representations of G. Given f 2 V, the Fourier transform fb of f is the complex valued function on Gb defined by fb(π) = Θπ(f). Plancherel measure is a measure ν on Gb having the property that Z f(1) = fb(π) dν(π); f 2 V: Gb This expression is called the Plancherel formula. A character Θπ is an example of an g g −1 invariant distribution on G:Θπ(f ) = Θπ(f) for all g 2 G, where f (x) = f(gxg ). The map f 7! f(1) is also an invariant distribution. Invariant distributions play an important role in harmonic analysis. Given an invariant distribution D, we can try to find a measure ν on G such that D b Z D(f) = fb(π) dνD(π); f 2 V: Gb Note that Plancherel measure is νD for D(f) = f(1). Harmonic analysis on locally compact 3 abelian groups and compact groups is well understood. Much work has been done for Lie groups, nilpotent, solvable, and semisimple Lie groups, etc. A fair amount is known for nilpotent and solvable p-adic groups. However, for reductive p-adic groups, there is very little known, and there are many open problems in harmonic analysis. For example, the characters of the irreducible representations of reductive groups over p-adic fields are not yet well understood. In this course, we study some basic results in the theory of admissible representations of reductive p-adic groups. These results play important roles in applications to auto- morphic forms and harmonic analysis. As the structure of reductive algebraic groups is not discussed here, many of the results will be stated and proved for p-adic general lin- ear groups. In these cases, the main ideas of the argument for arbitrary reductive p-adic groups is the same as that for general linear groups. 4 2. Valuations and local fields The references for this part are the books by Bachman and Cassels. Let F be a field. Definition. A valuation on F is a map j · j : F ! R≥0 such that, for x, y 2 F , jxj = 0 () x = 0 jxyj = jxjjyj jx + yj ≤ jxj + jyj c Definition. Two valuations j · j1 and j · j2 on F are said to be equivalent if j · j1 = j · j2 for some positive real number c. Definition. A valuation j · j is discrete if there exists δ > 0 such that 1 − δ < jaj < 1 + δ implies jaj = 1. That is, f log jaj j a 2 F × g is a discrete subgroup of the additive group R. Example: Let p be a fixed prime integer. Given x 2 Q×, there exist unique integers m, n and r such that m and n are relatively prime, p does not divide m or n, and x = prm=n. −r Set jxjp = p (and j0jp = 0). This defines the p-adic valuation on Q. This valuation is discrete. Observe that if x = prm=n and y = p`m0=n0, then prmn0 + p`m0n jx + yj = j j = jprmn0 + p`m0nj p nn0 p p −r −` ≤ maxfp ; p g = maxfjxjp; jyjpg: Furthermore, if jxjp 6= jyjp, then jx + yjp = maxfjxjp; jyjpg. Definition. A valuation j · j is non archimedean if j · j (is equivalent to) a valuation which satisfies the ultrametric inequality: jx + yj ≤ maxfjxj; jyjg, x; y 2 F . Otherwise j · j is archimedean. Example: If F is a finite field, then the only valuation on F is the trivial valuation.