Represents the restriction to $K$ of a (coherently) continued standard Harish-Chandra module.
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Represents the restriction to $K$ of a (coherently) continued standard Harish-Chandra module.
This is a parameter type like Tits elements; the important operations are modifying and comparing values, not storing additional data that facilitate methods. For that, auxiliary classes |SRK_context| and |KhatContext|, which has a role similar to |WeylGroup| with respect to |WeylElt|, will be used
For us a standard Harish-Chandra module is attached to 1) a real Cartan subgroup $H(R)=H(R)_c H(R)_s$, with $H(R)_c = H(R) K$ the compact part, and $H(R)_s$ (a vector group) the noncompact part; 2) a system ${im]$ of positive imaginary roots for $H(R)$ in $G$; 3) a system ${re}$ of positive real roots for $H(R)$ in $G$; and 4) a genuine character $$ of the rho-cover $ H(R)$ (here genuine means that $$ is in $X^*+$, so unless $ X^*$, the character does not descend to a character of $H(R)$).
Action of the real Weyl group preserves the meaning of these data.
(We said "attached to" rather than parametrized, as there are subtle identifications and relations, which are associated to the notions of being "standard" (rather than continued), "final", and "normalized").
In the Atlas picture, the Cartan and complete positive root system are always fixed, so one does not specify 2) and 3); instead the situation will be conjugated to one where the positive roots are the perpetual ones, and what changes is the strong involution $x$ representing the real form, and the position of $$ with respect to the simple coroots (it need not be dominant for all of them). As in the KGB module, strong involutions are represented by Tits group elements, the precise correspondence depending on a "base grading" stored elsewhere. In this class we will always assume that the involution $$ on $H$ is distinguished within its class, using $W$-conjugacy where necessary to make it so. This means that except for intermediate computational values, the Weyl group part of the Tits element can be replaced by an indication of the number |d_cartan| of the real Cartan we are considering (the Weyl group part is then the twisted involution for the distinguished involution in that class). The (left) torus part is explicitly represented as |d_fiberElt|; it implicitly determines the real form as in the Cartan class module, provided the central square class (which is stored elsewhere) is known.
Since we are interested only in HC modules restricted to $K$, we are interested only in the character $$ restricted to $ H(R)_c$. Characters of the compact group $H(R)_c$ are the same as algebraic characters of the complexification; that is
${H(R)_c}$ is identified with $X^* /(1-)X^*$.
At the level of the $$-cover, we get
($$ restricted to $K$) is identified with an element of the coset-quotient $(X^* + )/(1-) X^*$.
This is the information held in |d_lambda|. (The name $$ refers to the restriction of $$ to $ H(R)_c$.) This is of type |HCParam|, consisting of an integer vector and an integer respesenting a bit vector; the integer vector gives the non-torsion part of $(X^*+)/(1-)X^*$ on a basis of $(1/2)X^*$ held in |KhatContext|, and the bitvector gives the torsion part, via a basis also given there.
1.8.11
