Questions tagged [p-adic-groups]
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290 questions
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Iwahori spherical representations of GL(n) with no nonzero fixed vectors under the fixator of a panel of the affine building
Let $G$ be the group $GL(n,F)$, where $F$ is a p-adic field, and $I$ its standard Iwahori subgroup. Let $\pi$ be an irreducible smooth representation of $G$ with nonzero $I$-fixed vectors. It is well ...
- 21
5
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1
answer
134
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Regular elliptic elements are dense in p-adic division algebra
I'm trying to better understand the set $E$ of regular elliptic elements of $D^\times$, where $D$ is a finite dimensional central division algebra over a non-archimedean local field $F$.
For example, ...
- 208
9
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1
answer
223
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$p$-adic analytic pro-$p$ group satisfies a pro-$p$ identity?
Let $p$ be a prime. Let $w$ be an element of a free pro-$p$ group $F_r$ of finite rank $r\geq 2$. Then we say that a pro-$p$ group $G$ satisfies the pro-$p$ identity
$w$ if for every homomorphism $ f:...
- 667
2
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0
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57
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Hall-Littlewood polynomials for $n$-tuples that are not partitions
For some calculations related to the unramified principal series of ${\rm GL}(n)$ over a $p$-adic field, I need to compute Hall-Littlewood polynomials that are associated to $n$-tuples that are not ...
- 6,349
3
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1
answer
179
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Finiteness in Bernstein blocks
In his Cuspidal geometry of p-adic groups [J. Anal. Math. 47, 1–36 (1986)], Kazhdan uses a standard (?) result about representations of $p$-adic groups, which I will try to restate here.
Let $F$ be a $...
- 133
2
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0
answers
118
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Growth of invariants in mod-$p$ representations of $\mathrm{GL}_n$
Let $G$ be smooth admissible mod-$p$ representations of $\mathrm{GL}_n(\mathbb{Q}_p)$. Also suppose $\pi$ is an irreducible infinite-dimensional smooth admissible representation of $G$ over $\mathbb{F}...
6
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1
answer
126
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Intersection of integral points with a unipotent and its opposite
This is a follow-up to Does the bruhat decomposition induces decomposition on integral points (on an open cell)?
Given a split connected reductive group $G$ over a $p$-adic local field $F$ with ring ...
- 503
2
votes
0
answers
44
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Nonvanishing of Jacquet modules of principal series subquotients
Let $G$ be a connected reductive group over a $p$-adic field $F$. For simplicity, we assume $G$ to be split. Fix a Borel $B=TU$ with its Levi decomposition ($U$ unipotent radical, $T$ maximal torus). ...
- 649
1
vote
0
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75
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Cartan decomposition over a not-necessarily-discretely-valued field
Let $K$ be a valued field, and let $R$ be the valuation ring of $K$. Let $G$ be a split reductive group over $K$ and $T$ a maximal torus of $G$. On page 107 Berkvoich's book "Spectral theory and ...
- 63
2
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1
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130
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Extending $p$-adic smooth and locally constant functions
Let $G$ be $p$-adic group and let $G \rightarrow GL(V)$ be a representation. For example, $V$ is a quadratic $\mathbb{Q}_p$-space and $G$ is the associated orthogonal group.
Take a point $v \in V$, ...
- 81
3
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67
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Same Bruhat-Tits buildings for different groups
In dimension 1 it can happen, that two p-adic groups share the same Bruhat-Tits building as the latter are highly symmetric trees.
Can the same happen in higher dimensions as well?
If it happens, is ...
- 460
3
votes
0
answers
112
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The maximal $p$-abelian $p$-ramified extension of the cyclotomic $\mathbb{Z}_p$-extension
$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Cl{Cl}$[Reference: S. Lang, Cyclotomic Fields I and II, §2 chap 6]
Let $K$ be a number field. Suppose that $K$ contains the $p$-th roots of unity if $...
- 367
1
vote
0
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77
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Asymptotic behavior of Shalika germs near non-regular elements
Let $G$ be a connected reductive group over a $p$-adic field $F$. Let $T\subset G$ be a maximal torus. Fix a special maximal compact subgroup $K$ of $G(F)$ and for any closed subgroup $H\subset G(F)$ ...
- 649
2
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1
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209
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Question on the irreducibility of induced representations of $\mathrm{GL}_n$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\det{det}\DeclareMathOperator\Ind{Ind}$Let $F$ be a $p$-adic local field and $\chi_i$ be unramified characters of $F^{\times}$. For a integer $m$, ...
- 1,019
3
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1
answer
144
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Computation of modulus character of the mirabolic subgroup of $\operatorname{GL}_n$ using roots
$\DeclareMathOperator\GL{GL}$For parabolic subgroup of a general linear group or classical group $G$, we can compute its modulus character using the positive roots associated to them.
But it seems ...
- 1,019
2
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0
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119
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Question on the geometric lemma in $p$-adic classical groups
$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}\DeclareMathOperator\B{B} $My question is about the geometric lemma of $p$-adic classical groups not $\GL_n$. For $...
- 1,019
3
votes
1
answer
170
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Factoring out an element of a root subgroup to make a conjugation integral
Fix a nonarchimedean local field $L/\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, uniformizer $\varpi$, and residue field $k$. If I take a matrix
$$\begin{pmatrix} a & \varpi b \\ c & d \...
- 503
1
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0
answers
47
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Question on the modulus character of not parabolic subgroup
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$
Last time, I asked a question on the computation about modulus character of parabolic subgroup of symplectic group and LSpice gave me a nice ...
- 1,019
4
votes
1
answer
130
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Do parabolic inductions share a composition factor if and only if the inducing data are associate?
Let $F$ be a local field of characteristic zero and $G$ a connected reductive group over $F$. Let us call an inducing datum a triple $(P,M,\sigma)$, where $P$ is a parabolic subgroup of $G$, $M$ is a ...
- 387
13
votes
1
answer
289
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$p$-adic counterpart of W-algebra
Representation theory and geometry over $k((t))$ and $\mathbb{Q}_p$ have many similarities, and there are many similar constructions, usually motivated from the other side (say the study of affine ...
- 1,391
1
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0
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139
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Question on two types of Frobenius theorem in $p$-adic groups
Let $G$ be a $p$-adic classical group and let $P_0$ be a minimal parabolic subgroup of $G$. Let $P=MN$ be a
standard parabolic subgroup containing $P_0$. Let $\text{Ind}$ and $\text{Jac}$ be the ...
- 1,019
7
votes
1
answer
347
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$\mathbb{Z}$-homomorphism and $\mathbb{Z}_p$-homomorphism
$\newcommand{\cts}{\mathrm{cts}}$Thanks for your reading. Let $A,B$ be two $\mathbb{Z}_p$-modules, where $\mathbb{Z}_p$ is the $p$-adic integer ring. I have two questions.
Is $\mathrm{Hom}_{\mathbb{Z}...
- 319
1
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0
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48
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Genericity of local representation with a non-generic local A-parameter
Let $\pi$ be an irreducible smooth representation of a classical $p$-adic group. Suppose that $\pi$ has a local L-parameter associated to some non-generic local A-parameter $\psi$. Then I am wondering ...
- 1,019
2
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0
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71
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On the conjugation action on the first congruence subgroup of special linear group over $p$-adic fields
Let $\mathcal{O}_{L}$ be the ring of integers of a finite extension $L$ of $p$-adic number fields $\mathbb{Q}_{p}$ where $p$ is an odd prime. Let $\mathfrak{m}_{L}$ be the maximal ideal of $\mathcal{O}...
- 667
3
votes
0
answers
91
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Question on the genericity of unramified representation
Let $F$ be a $p$-adic local field and $W$ be a 2n-dimensional symplectic space over $F$. Let $G_n$ be the isometry group of $W$ and $B_n$ be the Borel subgroup of $G_n$. Then the maximal torus $T_n$ ...
- 1,019
2
votes
1
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166
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Integral over the space of $p$-adic matrices
$\DeclareMathOperator\Mat{Mat}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers with a uniformizer $\pi$. Let $|\cdot|\colon \mathbb{F}\to \mathbb{R}$ be ...
- 21.8k
3
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0
answers
65
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One parameter subgroups of reductive algebraic groups
If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...
- 63
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0
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110
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Hecke algebra $\mathcal{H}(K_1\backslash \mathrm{GL}_n(\mathbb{F})/K_1)$
$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers, and let $\frak{m}$ be its maximal ideal Let $\GL_n(\mathcal{O})$ be the group ...
- 21.8k
4
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1
answer
170
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Representations of $\mathrm{GL}_n(\mathcal{O})$ in functions on Grassmannians
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers.
The natural representation of the group $\GL_n(\...
- 21.8k
2
votes
0
answers
76
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Simplest way to classify reducibility of principal series for $p$-adic $\mathrm{SL}_2$
Let $F$ be a $p$-adic field and $G_1=\mathrm{SL}_2(F)\subset \mathrm{GL}_2(F)=G$. For simplicity, we assume $p>2$. Denote by $|\cdot|$ the normalized absolute value on $F$. Here I shall focus on ...
- 649
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0
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377
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Subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$
Assume that $n \geq 3$. Is there a subgroup $H \leq {\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$ whose order is a power of $2$ so that
$\bullet$ $\det : H \to (\mathbb{Z}/2^{n} \mathbb{Z})^{\times}$ is ...
- 20.4k
2
votes
0
answers
76
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Simple question on the genericity of induced representation
$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}$
Let $F$ be a $p$-adic field and $\Sp(2n)$ symplectic group over 2n dimensional symplectic space over $F$.
Let $B=...
- 1,019
1
vote
0
answers
98
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Question on the unramified representation
$\DeclareMathOperator\GL{GL}$Let $F$ be a $p$-adic field and $\chi$ be an unramified character of $\GL_1(F)$.
Consider an induced representation $\pi$ of $\GL_2(F)$ induced from the character $\chi|\...
- 1,019
4
votes
0
answers
213
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Gelfand-Kazhdan criterion, exposition by Paul Garrett
Here is Paul Garrett's exposition on the Gelfand-Kazhdan Criterion.
In page 4 of the exposition, he showed the following lemma.
Lemma (Page 4): Let $B, t, S$ be as above and for $\alpha, \beta$ in $...
- 43
2
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1
answer
201
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Orbital integrals of $\operatorname{SL}_2$ and the fundamental lemma
$\DeclareMathOperator\SL{SL}$When I was checking some orbital integral computations of Sally-Shalika's The Plancherel Formula for $\SL_2$ over a Local Field, Proceedings of the National Academy of ...
- 649
1
vote
1
answer
89
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Compact subgroups of a linear group over non-Archimedean local field
$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers.
Is it true that any compact subgroup of $\GL_n(\mathbb{F})$ is conjugated to ...
- 21.8k
1
vote
0
answers
113
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Jacquet module for characteristic p
Let $F$ be a finite extension of $\mathbb{Q}_p$ and let $G=\operatorname{GL}_2(F)$ (or $\operatorname{GL}_n$ or any reductive group). I consider smooth representations of $G$ in characteristic $p$. ...
- 61
4
votes
1
answer
170
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Partition of unity for analytic manifolds over non-Archimedean local fields
I am looking for a reference to the following fact which, I hope, is correct.
Let $X$ be a compact analytic manifold over a non-Archimedean local field. Let
$X=\cup_\alpha U_\alpha$ be a finite open ...
- 21.8k
5
votes
0
answers
132
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Is $\mathbf{C}_p(X)$ self-dual?
Let $X$ be a set. Consider $\mathbf{Q}_p$ and $\mathbf{Z}_p$ as the $p$-adic numbers and $p$-adic integers, respectively. For any finite subset $F \subseteq X$, one can construct the topological ...
- 1,485
2
votes
1
answer
438
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Question on the modulus character of classical p-adic group
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$It is well known for the formula of the computation of modulus character of general linear groups. For example, for the standard Borel subgroup $...
- 1,019
2
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0
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106
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Hereditarily just-infinite pro-$2$ groups
An infinite profinite group $G$ is called just-infinite if all non-trivial closed normal subgroups of $G$ have finite index. A profinite group is called hereditarily just-infinite if every open ...
- 667
3
votes
1
answer
327
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Pro-unipotent radical (in Bruhat-Tits) vs. unipotent radical (and reference request)
I'm a beginner in Bruhat-Tits theory, and the following phenomenon makes me puzzled.
So let $F$ be a $p$-adic field, with $\mathfrak{o}\supset \mathfrak{p}$ its ring of integers and maximal ideal, and ...
- 649
2
votes
0
answers
165
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Prime-to-$p$ quotients of ${\rm PSL}_{2}(\mathbb{Z}_{p})$
Let $p$ be a prime and $\mathbb{Z}_p$ the ring of $p$-adic integers. Let ${\rm PSL}_{2}(\mathbb{Z}_{p})={\rm SL}_{2}(\mathbb{Z}_{p})/\{\pm 1\}$ be the projective special linear group over $\mathbb{Z}...
- 667
2
votes
0
answers
169
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$\mathrm{Ext}^i(\pi_1, \pi_2)\neq0$ implies same central character
If $\pi_1$ and $\pi_2$ are two smooth admissible representations of $\operatorname{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$ with central characters. I want to prove that if $\pi_1$ has ...
user515481
1
vote
1
answer
194
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Non-split extension of representations of $\mathrm{GL}_2$ and $\mathrm{Hom}$
Let $0\to V_1\to V\to V_1\to0$ be a sequence of representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$, where $V_1$ is irreducible, smooth and admissible. Assume that this ...
user515481
5
votes
2
answers
533
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Non-trivial extension of representations have same central character
Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations ...
user515481
3
votes
0
answers
107
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(Non)complete abelian groups in the “transfinite p-adic topology”
For an abelian group $A,$ a prime $p$ and an ordinal $\alpha,$ we recursively define $p^\alpha A$ as a subgroup of $A$ such that $p^0A=A,$
$$p^{\alpha+1}A=p(p^\alpha A) \hspace{5mm} \text{and} \...
- 1,484
1
vote
0
answers
102
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Does $\tilde{\mathrm E}_{6,3}^{(2)6}$ exist over a p-adic field?
Does a form of $\tilde{\mathrm E}_6^{(2)}$ with this Satake-Tits diagram exist over a p-adic field?
- 2,773
5
votes
1
answer
220
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Two different local Langlands parameters for quadratic extension
Let $E/F$ be a quadratic extension of nonarchimedean local field, and let $G$ be a reductive group over $E$, and $G(E)$ the $E$-points of $G$. (For my question, one may just assume $G=\operatorname{GL}...
- 833
1
vote
0
answers
140
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Explicit construction of $T$-orbits of generic characters of unitary groups
Let $F$ be a $p$-adic field. Let $E$ be a quadratic extension of $F$ and $G$ be a quasi-split unitary group $U(2n)$ or $U(2n+1)$ over with respect to $E/F$. Let $N_{E/F}$ be a norm map.
Let $B=TU$ be ...
- 1,019