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5 Fig. 3. 5 3 for a = 0 (solid), a = 1 (dotted), and a = 2 (dashed) 5. Proofs In this section we prove Theorems 3 and 4. First we remark that the unitary case has been studied in the work of Nagao and Wadati [24, 25], but we reproduce the proofs here for completeness and also to show that the hypothesis a > −1 is, in a certain sense, unnecessary. Also we remark that Forrester and Nagao [23] have studied the hard edge correlations directly, using skew-orthogonal polynomial expressions for the matrix kernels KR1 , KR4 , but their results apply only when the parameters a, b are strictly positive, Matrix Ensembles Associated to Symmetric Spaces 49 and in view of the application to symmetric spaces this restriction is unacceptable (see Table 1).

The KAK decomposition has an integral counterpart. Proposition 2 (Weyl’s integration formula). There is a measure d ν¯ (a) on A such that, for any f ∈ C(G), f (g)dg = G f (k1 ak2 )d ν¯ (a)dk2 dk1 . ) Denote by + the set of positive roots of the symmetric Lie algebra (g, ω), and by mα the multiplicity of a positive root α ∈ + . Then d ν¯ (a) ∝ | sin α( )|mα da =: α∈ say, where ( )da, (45) + is chosen so a = exp(i ). (With the notation above, we write i = log(a). ) Now recall that the (positive) roots of (g, ω) are certain non-zero real-valued linear functionals on a (in fact one should speak about the roots which are positive with respect to a fixed Weyl chamber in a).

Math. Pura Appl. 4, 209–256 (1927) 4. : ´ Sur certaines formes riemannienes remarquables des g´eom´etries a groupe fondamental 5. : ´ simple. Ann. Sci. Ecole Norm. Sup. 44, 354–467 (1927) 6. : A method of integration over matrix variables. II. J. Phys. A 14(3), 579–586 (1981) 7. : Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52(12), 1491– 1552 (1999) 8. : Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory.