There are two critical components here in establishing your leadership value as you transition to the new culture: what you do to manage yourself and adjust to the new culture, and what you do to demonstrate to your team that you are committed, “fit in” and add value.
The former – self-management – has to do with monitoring your thoughts and reactions and adjusting or reframing them to support your best actions for outcomes. This considers elements like stress management, adequate rest and nutrition and a support system that are especially helpful during times of change and transition to minimize negative thinking and emotional reactions that can flare during times of stress and change.
Next, understand that external/outward leadership behaviors most valued these days, especially given the generational differences in the workplace are: honesty/transparency, open problem solving, willingness to/doing things differently (innovation) and take risk, and a focus on results. These are core attributes in addition to an expectation of strong business acumen, technical capabilities, and strong interpersonal skills.
So, to help you quickly adjust and demonstrate value, I suggest you focus first on understanding the expectations and priorities of your boss, so you can clearly align your efforts, team and resources in that direction. (Should team members go direct to him, they would get the same information, so there is no disparity in focus and direction to create political dynamics. Also, you can communicate that as you are now here – they can come to you vs. your boss, assuming your boss agrees and will support redirecting them to you if they keep coming to him.) Next, get to know each one of your team members – their strengths, interests and concerns. This will start a relationship of openness and trust from which you can build on and better collaborate. Understanding individual perspectives, capabilities and interests will enable you to better utilize and support your team. This demonstrates caring – and employees that feel their boss sincerely cares about them are more engaged and productive (and will come to you verses go around you to your boss).
Lastly, reframe open challenges to your direction not as a threat or disregard, but an as opportunity for you to refine your position or purpose, better educate/inform and influence others, and to expand your own thinking to more quickly correct in action, should you find value in the challenge. I would inquire as to why they are challenging – for example, what are they thinking, concerned about or suggesting instead? If the issue is about the way they are challenging you – i.e., with disrespectful tone or comments – this becomes a private coaching issue to help them develop in giving meaningful feedback and respectfully inquiring or pushing back.
Best of luck.