Recursion
Collected so far
Each finished deformation adds its lowest K-types here, times the product of the coefficients on its path from the root. A cached call adds its whole stored result. Rows cancel as the computation proceeds.
Atlas · signatures of Hermitian forms
The c-invariant form on the spherical principal series I(kρ) of SL(2,ℝ), Sp(4,ℝ) or split G2. Deform ν = tkρ from t = 1 to 0. At each reducibility point the form changes, and atlas writes the change as a combination of standard modules with smaller ν. Each of those is deformed in turn, until everything is tempered.
Coefficients lie in 𝕎 = ℤ[s]/(s2−1), where s records a negative sign. Every cq is a multiple of (1−s). rp(p) is the set of reducibility points t ∈ (0,1] along the ray tν. LKT(p) is the lowest K-type part, which is the form at ν = 0. A tempered representation is named by its lowest K-type x=… [λ], as atlas prints it. When t = 1 is itself a reducibility point, atlas computes the form on I(kρ+ε). The recursion is Rep_table::deformation, memoized as in atlas, and all data were computed in atlas.
Each finished deformation adds its lowest K-types here, times the product of the coefficients on its path from the root. A cached call adds its whole stored result. Rows cancel as the computation proceeds.
The Cartan column is atlas's number for the Cartan class of the tempered term. Cartan 0 is the compact Cartan, so those terms are discrete series. The last column is the ordinary Hermitian form, hermitian_form_std(p). In the equal rank case it is the c-form twisted by an element of K.
c_form_std(p) with :