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Let $X\left( t \right)$ be a continuous gaussian stochastic process with zero mean $E\left[ X\left( t \right) \right]= 0$, the index set $ \left[ 0,T \right]$, $t\in \left[ 0,T \right]$. One chooses the time instants $t_{1}<t_{2}<.....<t...

Are there $T_2$-spaces $X, Y$, each having more than $1$ point, such that every continuous map $f:X\to Y$ is constant, and every continuous map $g:Y\to X$ is constant?

Let $\pi$ be an irreducible generic representation of $\operatorname{GL}_n(\mathbb{R})$. Is it known whether the restriction of $\pi$ to the mirabolic subgroup $P_n(\mathbb{R})$ is indecomposable? If not, any ideas how it can be shown? Any help would...

Apologies in advance for the long setup and question.
Let $L \to X$ be a line bundle. We may take its frame bundle $p \colon Fr(L) \to X$, a $\mathbb{G}_m$-torsor. We have
$$ p_*\mathcal{O}_{Fr(L)} = \bigoplus_{k \in \mathbb{Z}} {L}^{\otimes (-k)}. $$