Explicit Formulas for the Affine Satake Isomorphism

Definitions and mathematical statement

Let $G$ be reductive, let $G_{\mathrm{aff}}$ denote the corresponding affine Kac--Moody group intended by the source, and let $G_{\mathrm{aff}}^\vee$ denote its affine Langlands dual. For a specified tensor category $\operatorname{Rep}(G_{\mathrm{aff}}^\vee)$, its representation Grothendieck ring $K_0(\operatorname{Rep}(G_{\mathrm{aff}}^\vee))$ has generators $[V]$, relations $[V]=[V']+[V'']$ for short exact sequences $0\to V'\to V\to V''\to0$, and multiplication induced by tensor product. The source writes this ring as $K(\operatorname{Rep}(G_{\mathrm{aff}}^\vee))$. Its proposed affine Satake setting is schematically \[\mathcal S_{\mathrm{aff}}:\mathcal H_{\mathrm{aff}}\xrightarrow{\ \sim\ }\widehat{K_0(\operatorname{Rep}(G_{\mathrm{aff}}^\vee))\otimes_{{\mathbb{Z}}}{\mathbb{C}}},\] where the hat denotes the source's unspecified completion and $\mathcal H_{\mathrm{aff}}$ denotes what it calls the affine Hecke algebra. The question is whether this Satake isomorphism admits an explicit formula analogous to Lusztig's explicit $q$-analogue formula. The source supplies neither the analogue itself nor its coefficient/basis normalization, so no particular coefficient identity is asserted here. The algebra on the left must be clarified: the ordinary full Iwahori affine Hecke algebra is generally noncommutative, whereas a representation ring of this kind is commutative; a spherical algebra or another specified Satake algebra is therefore essential to the intended reading, rather than an automatic identification with every affine Hecke algebra. The construction of $G_{\mathrm{aff}}$, the dual root datum, the representation category, level conditions, completion, base field, and role of $q$ are not specified in the extract. The supplied 2026-08-21 triage attributes the proposed analogue to Braverman, describes its affine or double-affine Grassmannian program as conjectural in subsequent papers, and labels the general formula open.

Short English statement

Ask for an explicit Lusztig-type $q$-analogue formula describing the intended affine Satake map. The extract does not fix the relevant Hecke algebra, representation category, completion, or coefficient normalization, so these must be recovered before a particular formula can be stated.

Sources

• Local UnsolvedMath registry, permanent ID 30000760, source problem catalogue entry, Explicit Formulas for the Affine Satake Isomorphism, proposed year 2007. No individual proposer is supplied; the literature assessment calls the analogue Braverman's conjecture.
• Supplied report citation: Algebraic Groups (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/22, pp. 1191--1242: https://doi.org/10.4171/owr/2007/22.
• Supplied literature label: Braverman--Finkelberg, double affine Grassmannian I, https://arxiv.org/abs/0711.2083.
• Supplied literature label: Braverman--Finkelberg, double affine Grassmannian II, https://arxiv.org/abs/0908.3390.