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Howe Pairs, Supersymmetry, and Ratios of Random Characteristic HOWE PAIRS, SUPERSYMMETRY, AND RATIOS OF RANDOM CHARACTERISTIC POLYNOMIALS FOR THE UNITARY GROUPS UN by J.B. Conrey, D.W. Farmer & M.R. Zirnbauer Abstract. — For the classical compact Lie groups K UN the autocorrelation functions of ratios of characteristic polynomials (z,w) Det(z≡ k)/Det(w k) are studied with k K as random variable. Basic to our treatment7→ is a property− shared− by the spinor repre- sentation∈ of the spin group with the Shale-Weil representation of the metaplectic group: in both cases the character is the analytic square root of a determinant or the reciprocal thereof. By combining this fact with Howe’s theory of supersymmetric dual pairs (g,K), we express the K-Haar average product of p ratios of characteristic polynomials and q conjugate ratios as a character χ which is associated with an irreducible representation of χ the Lie superalgebra g = gln n for n = p+q. The primitive character is shown to extend | to an analytic radial section of a real supermanifold related to gln n , and is computed by | invoking Berezin’s description of the radial parts of Laplace-Casimir operators for gln n . The final result for χ looks like a natural transcription of the Weyl character formula| to the context of highest-weight representations of Lie supergroups. While several other works have recently reproduced our results in the stable range N max(p,q), the present approach covers the full range of matrix dimensions N N. ≥To make this paper accessible to the non-expert reader, we have included a chapter∈ containing the required background material from superanalysis. arXiv:math-ph/0511024v2 9 Sep 2007 Contents 1.Introduction ..................................... ......... 3 1.1.Statementofresult .............................. .... 4 1.1.1.Howeduality .................................. .. 5 1.1.2. Representation-theoretic formulation . .... 6 1.1.3.Compactsector ................................ .. 7 1.1.4.Non-compactsector ............................ .. 8 1.1.5.Stableversusnon-stablerange .. ......... ....... .. 8 1.1.6.Otherapproaches .............................. .. 9 1.2. Somebasic definitions of superanalysis .. .. .. .. .. ... 9 1.3.Thegoodobjecttoconsider ........................ .. 12 1.4.Outlineofstrategy .............................. ...... 13 2. Autocorrelation function of ratios as a primitive character ...... 15 2.1. Clifford algebra, spinor module, and character formula .... 15 2 J.B. CONREY, D.W. FARMER & M.R. ZIRNBAUER 2.1.1.Spinorrepresentation ......................... ... 15 2.1.2. GL(V) characterformula .......................... 16 2.2.Weylalgebraandoscillatormodule ........... ...... ... 18 2.3.Oscillatorcharacter ............................ ...... 19 2.4.Entersupersymmetry ............................. ... 20 2.5.Supersymmetricframework ........................ .. 22 2.6.Howedualityandaconsequence ..................... 24 2.7. Weight expansion of χ ................................ 26 3. Background material from superanalysis .. ............ ........ 29 3.1. The vector bundle underlying a classical Lie supergroup .. 30 3.2. g-actiononaLiesupergroup .......................... 31 3.2.1. The action of g0 .................................. 31 3.2.2.Grassmannenvelope ............................ .. 32 3.2.3. The action of g1 .................................. 32 3.2.4.Summary ...................................... .. 35 3.3. What’s a representation in the supergroup setting? . ..... 35 3.4. What’s a character of G? .............................. 36 3.5. χ is an eigenfunction of all Laplace-Casimir operators . 37 3.6.Radialfunctions ................................ ...... 39 3.7. Block diagonalization of supermatrices .. .... 40 3.8. Radial parts of the Laplacians for Um n .................. 41 4. Determination of the character χ ............................| 42 4.1.Thegoodrealstructure ........................... ..... 43 4.1.1. GR-principalbundle .............................. 44 4.1.2. Real form of gl(U) .............................. 45 4.1.3.Liesupergroupstructure ....................... ... 45 4.2. Extending the character χ ............................ 47 4.3. All Casimir invariants vanish on Vλ .................... 49 4.4.Maximaltorus ................................... ... 51 4.5. W-invariance ........................................ 53 4.6. Radial differential equations for χ ...................... 53 4.7.Proofofthemaintheorem .......................... .. 55 4.7.1.Uniquenessofthesolution ...................... .. 55 4.7.2.Explicitsolution ............................. ..... 58 4.8.ProofofCor.1.2 ................................. ... 62 5.Proofoftheextendedcharacter formula .. ............. ....... 63 s 5.1. More about the semigroup H(V0 ) ...................... 64 s s A 5.2. Representation of H(V0 ) and U(V0 ) on V .............. 65 5.3.Combiningtherepresentations .. .................. .... 71 5.4.ProofofProp.4.4 ................................ .... 72 References ......................................... ......... 75 HOWE PAIRS, SUPERSYMMETRY, AND RATIOS 3 1. Introduction Let K be one of the compact Lie groups UN ,ON , or USpN , and let K be equipped with Haar measure dk normalized by K dk = 1. We are interested in products of ratios of characteristic polynomials Det(z k), averaged with respect to the probability R measure dk . Our goal is to derive exact expressions− for such autocorrelation functions. Quantities of a similar kind, for Gaussian distributed Hermitian matrices, were first computed some time ago in the large-N limit by Andreev and Simons [1], using an elaboration of the supersymmetry method pioneered by Wegner and Efetov. Then, in 1996-97, one of us developed a variant [21] of the Wegner-Efetov method to handle the case of Haar-ensemble averages for the compact Lie groups above. With guid- ance from a perceptive remark of Sarnak [19], it soon transpired that the mathematical foundation of that variant had existed in classical invariant theory for decades, and is known there as Howe duality. The main ideas – namely, Howe dual pairs acting in the spinor-oscillator module, the formula for the spinor-oscillator character, and the fact that the K-Haar autocorrelation can be regarded as the character associated with a highest-weight representation of some Lie superalgebra g – were presented at the MSRI workshop on ”Random matrices and their Applications” (Berkeley, June 1999). The second step of that original approach was to express the primitive g-character as a co-adjoint orbit integral (actually, a Berezin superintegral) ´ala Kirillov, and finally to compute this integral by a super version of the localization principle for equivariantly closed differential forms [5]. While this last step works in the so-called stable range, i.e., for large enough N, the superintegral becomes too subtle an object to handle for small values of N and, in any case, this type of analysis has not yet been perfected to the stage of yielding mathematical theorems. In the present paper, we replace it by a combination of algebraic and analytical techniques avoiding superintegration. While this research was initially motivated by quantum chaos and the physics of dis- ordered systems, the main motivationfor bringing it to completion comes from analytic number theory. In that area, random matrices have been a recurrent and growing theme ever since Montgomery conjectured the pair correlation function of the non-trivial ze- roes of the Riemann zeta function to be asymptotic to Dyson’s pair correlation function for the Gaussian Unitary Ensemble (GUE) and since Odlyzko’s high-precision numer- ical study demonstrated asymptotic agreement of the distribution of spacings between the Riemann zeroes with the spacing distribution predicted from GUE. The conjectured relation between the zeroes of zeta functions and eigenvalues of ran- dom matrices was much clarified by the work of Katz and Sarnak [16], which shifted the emphasis from Riemann zeta to families of L-functions. Carried out in the setting of L-functions over finite fields, this work explained how a limit procedure for fam- ilies leads to equidistribution on one of the classical compact Lie groups SUN , UN , SO2N+1 , USpN , or SO2N , the group being determined by the symmetry of the family. Keating and Snaith made the proposal that the connection between L-functions and random matrices should carry over to the case of global L-functions. In particular, they conjectured [17] that the critical value distribution of the Riemann zeta function (or, more precisely, a certain universal part thereof, which is hard to access by number 4 J.B. CONREY, D.W. FARMER & M.R. ZIRNBAUER theory methods) can be modelled by the characteristic polynomial of a random matrix u UN . A sizable number of precise generalizations of this proposal have since been made,∈ although most of the theory to date remains at the conjectural level. In a companion paper [9] we present various conjectures that are suggested for L- functions by the Keating-Snaith conjecture and the results of our work. The present paper is solely concerned with the random-matrix aspect of this unfolding story; the reader is referred to our second paper for the related details concerning L-functions. 1.1. Statement of result. — We now define the quantity to be studied in this paper. Postponing the cases of the compact Lie groups K = ON and K = USp2N to another publication we focus on the case of K = UN and, fixing a pair of positive integers p and q, consider for u UN the product of ratios of characteristic polynomials
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