The scalar product formula for parahoric Deligne–Lusztig induction

Fix a character $\theta$ and parahoric data $(\mathbb{T}_r,\mathbb{B}_r)$, and let $(\theta',\mathbb{T}'_r,\mathbb{B}'_r)$ be arbitrary corresponding data. The scalar product is taken in the representation ring of the relevant finite reductive quotient, with the Weyl-group sum understood as in the scalar product formula.

The scalar product conjecture. For all $(\theta',\mathbb{T}'_r,\mathbb{B}'_r)$, one has

$$\left\langle R_{\mathbb{T}_r,\mathbb{B}_r}^{\mathbb{G}_r}(\theta),R_{\mathbb{T}'_r,\mathbb{B}'_r}^{\mathbb{G}_r}(\theta')\right\rangle=\sum_{w\in W_{\mathbb{G}_r}(\mathbb{T}_r,\mathbb{T}'_r)(\mathbb{F}_q)}\left\langle\theta,{}^w\theta'\right\rangle.$$

Establishing this formula in the parahoric setting is fundamental for comparing these representations with regular supercuspidal representations and for constructing associated $L$-packets; its resolution is not supplied in the source.