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Math. Phys. : Weyl’s character formula for non-connected Lie groups and orbital theory for twisted affine Lie algebras. J. Funct. Anal.

4 The root system A0 λ+ρwc ,k+h∨ (H0 ) Fwc (H0 ) . 5. We have to distinguish two cases. First, let us suppose that for any simple root α ∈ , the roots α and σc (α) are not connected in the Dynkin diagram of . In this case, one can easily show that σc (eα ) = eσc (α) , so that s(α) = 1 for all real roots α ∈ re . 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 .

The basic levels of non-simply connected Lie groups have been calculated in [T]. See also [FSS] for a list of the root systems σ for general automorphisms σ of the Dynkin diagram of . The notation for affine root systems in the table below is the same as in [K]. G SLn c G = G/ c (1) n ≥ 2 Zr Spin2n+1 n ≥ 2 Z2 kb SO2n+1 An/r−1 if r = n n(n−1) k r2 ∅ if r = n ∈Z 1 Sp4n n ≥ 1 Z2 1 Sp4n+2 n ≥ 1 Z2 2 Spin4n n ≥ 2 Z02 Spin4n n ≥ 2 Z± 2 SO4n σc (1) smallest k with An−1 1 1 if n even (1) Bn (1) C2n (1) C2n+1 (1) D2n (1) D2n (2) A2(n−1) (2) A2n (1) Cn (1) C2n−2 (1) Bn 2 if n odd (1) (1) (1) (1) Spin4n+2 n ≥ 2 Z2 SO4n+2 1 D2n+1 C2n−1 Spin4n+2 n ≥ 2 Z4 PSO4n+2 4 D2n+1 Cn (1) G2 (1) F4 E6 Z3 3 E6 E7 Z2 2 E7 (1) (1) 580 R.

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