# A question about mod $p$ local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$

In the mod $$p$$ local Langlands correspondence for $$\mathrm{GL}_{2}(\mathbb{Q}_{p})$$, the irreducible supercuspidal representation $$\left(\mathrm{ind}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}\mathrm{Sym}^{r}\overline{\mathbb{F}}_{p}^{2}\right)/T$$ of $$\mathrm{GL}_{2}(\mathbb{Q}_{p})$$ is mapped to the irreducible Galois representation $$\mathrm{ind}(\omega_{2}^{r+1})$$ for $$r\in\lbrace 0,\ldots,p-1\rbrace$$. See Definition 1.1 in

https://www.imo.universite-paris-saclay.fr/~breuil/PUBLICATIONS/GL2%28Qp%29II.pdf (with $$\eta=1$$).

However, I notice that this is not compatible with the local class field theory: on $$\mathbb{Z}_{p}^{\times}$$, the central character of the supercuspidal representation is the $$r$$-th power map after going modulo $$p$$, while the determinant character of the Galois representation is the $$(r+1)$$-th power map.

So is it defined this way to satisfy some other compatibilities (for example, reduction modulo $$p$$ of the $$p$$-adic local Langlands for $$\mathrm{GL}_{2}(\mathbb{Q}_{p})$$)?

You seem to be expecting that mod $$p$$ local Langlands should satisfy the same compatibilities as "conventional" local Langlands (for smooth representations of $$GL_2(\mathbf{Q}_p)$$ and $$WD(\mathbf{Q}_p)$$ with coefficients in $$\mathbf{C}$$).
However, before you can even talk about reduction mod $$p$$, you need to check that the coefficients can be descended from $$\mathbf{C}$$ to a number field. I.e., if $$\pi$$ is a smooth irred rep of $$GL_n(\mathbf{Q}_p)$$ on an $$L$$-vector space, where $$L$$ is some subfield of $$\mathbf{C}$$, then is its Langlands parameter $$\phi_{\pi}$$ an $$L$$-valued Weil-Deligne rep? The answer, annoyingly, is "no": if $$n$$ is even, you have to add $$\sqrt{p}$$ to $$L$$ in order to get this to work. So it's common to re-normalise by twisting the correspondence for $$GL_n$$ by $$|\det|^{(n-1)/2}$$; this makes it compatible with coefficient fields, and also works better for local-global compatibility.
It's this same shift which you are seeing in the mod $$p$$ theory, and here it's completely impossible to get rid of, even if you extend the coefficient fields as much as you like: the character $$|\cdot|^{1/2}$$ of $$\mathbf{Q}_p^\times$$ is not $$p$$-adically unitary, so there is no way you can reduce it mod $$p$$.