By Xiangdong Xie

We determine a Bowen kind pressure theorem for the elemental crew of a noncompacthyperbolic manifold of finite quantity (with size at the least 3).

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For any root α ∈ let us denote by ασc its restriction to the subspace hσc ⊂ h. Then the set {mα ασc | α ∈ re } is the set of real roots of an affine root system which we will denote by σc . Now suppose that there exists some α ∈ such that α = σc (α) are not orthogonal. Since we have excluded the case that σc is the order n + 1 automorphism of the extended Dynkin diagram of An , we can assume that σc2 (α) = α. 1 for the case of finite root systems). Looking at the explicit form of the automorphism σc , we see that if there exists a simple root α such that α = σc (α) are not orthogonal, then α = σc (α) for all simple roots α of .

In both cases, the smallest positive imaginary root of σc is given by δσc = ord(σc )δ, where δ denotes the smallest positive imaginary root of . We will list the types of σc in the end of this paper. Now, using (1 + x)(1 − x) = (1 − x 2 ), we can write 1 − s(α)e−2πimα α(H0 ) = [α]∈( re )σc + 1 − e−2πα(H0 ) α∈ re σc + From this, we see that up to the factor f (q) = ord(wc ) ∞ −j q n )dim(hj ) n=1 j =1 (1 − ∞ ord(σc )n )mult(nδσc ) n=1 (1 − q , the function Fwc can be identified with the Kac-Weyl denominator corresponding to the affine root system σc .

Spherical functions on affine Lie groups. Duke Math. J. : Representations of affine Lie algebras, parabolic differential equations, and Lam´e functions. Duke Math. J. : Principal bundles over elliptic curves. : Holomorphic principal bundles over elliptic curves. II. The parabolic construction. J. Differential Geom. : Some automorphisms of generalized Kac-Moody algebras. J. : The action of Outer Automorphisms on Bundles of Chiral Blocks. Comm. Math. Phys. : From Dynkin diagram symmetries to fixed point structures.